![International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 4 Issue 4, June 2020
@ IJTSRD | Unique Paper ID – IJTSRD31139
A Classification of Groups
Ezenwobodo Somkene Samuel
Department of Mathematics, Nwafor Orizu College
ABSTRACT
Here we classified groups of order less than or equal to 15. We proved that
there is only one group of order prime up to isomorphism, and that all
groups of order prime (P) are abelian groups. This covers groups of order
2,3,5,7,11,13….Again we were able to prove that there are up to
isomorphism only two groups of order 2p, where p is prime and p
this is ܼଶ ≅ ܼଶ x ܼ. (Where Z represents cyclic group), and
dihedral group of the p-gon). This covers groups of order 6, 10, 14….. And
we proved that up to isomorphism there are only two groups of order P
And these are ܼమ and ܼ ݔ ܼ. This covers groups of order 4, 9…..Groups
of order P3 was also dealt with, and we proved that there are up to
isomorphism five groups of order P3. Whi
ܼ, ܦయ and ܳయ. This covers for groups of order 8… Sylow’s theorem was
used to classify groups of order ݍ, where p and q are two distinct primes.
And there is only one group of such order up to isomor
≅ ܼ ݔ ܼ. This covers groups of order 15… Sylow’s theorem was also used
to classify groups of order ଶ
ݍ and there are only two Abelian groups of
such order which are Zp
2
q and ܼ ݔ ܼ ݔ
groups of order one are the trivial groups. And all groups of order 1 are
abelian because the trivial subgroup of any group is a normal subgroup of
that group.
KEYWORDS: Abelian, cyclic, isomorphism, order, prime.
2010 Mathematics Subject Classification:
INTRODUCTION
The knowledge of Lagrange theorem and Sylow’s theorem
are important tools in the classification of groups. The
sylow’s first theorem helps us to present a group order in
the form of
ݍ where q doesn’t divide p. Lagrange
theorem helps us to know the possible divisors of a group.
Again the knowledge of the centre of a group, normal
subgroups contained in a group as well as direct product
of groups helps us to classify whether a group is abelian or
non-abelian. Using the notation of Gorenstein [5], any
finite Abelian group G is Isomorphic to a direct product of
cyclic groups of prime–power order. Berkovich [1], added
that this decomposition for ܩ will have the same number
of non-trivial factors of each other. For example; Z
Z3, Z12 ≅ Z3 × Z4. The knowledge of normal subgroup is an
indispensable tool in the study of group classification. It
will aid to differentiate abelian and non-abelian groups. A
normal subgroup is a subgroup that is invariant
conjugation by members of the group of which it is a part.
In other words, a subgroup ܰ of the group
if and only if ݃݊݃ିଵ
∈ ܰ for all ݃ ∈ ݀݊ܽ ܩ
as ܰ ⊲ .ܩ Evariste Galois was the first to realize the
significance of normal subgroups. Dummit [11], normal
subgroups are imperative because they (and only they)
can be used to create quotient groups of the given group.
Fraleigh[12], the normal subgroup of G are specifical
kernels of group homomorphisms with domain G, which
implies that they can be used to internally classify those
homomorphisms.
International Journal of Trend in Scientific Research and Development (IJTSRD)
2020 Available Online: www.ijtsrd.com e-ISSN: 2456
31139 | Volume – 4 | Issue – 4 | May-June 2020
f Groups of Small Order upto Isomorphism
Ezenwobodo Somkene Samuel
f Mathematics, Nwafor Orizu College of Education, Nsugbe, Nigeria
Here we classified groups of order less than or equal to 15. We proved that
there is only one group of order prime up to isomorphism, and that all
groups of order prime (P) are abelian groups. This covers groups of order
o prove that there are up to
isomorphism only two groups of order 2p, where p is prime and p≥3, and
. (Where Z represents cyclic group), and ܦ (the
gon). This covers groups of order 6, 10, 14….. And
roved that up to isomorphism there are only two groups of order P2.
. This covers groups of order 4, 9…..Groups
was also dealt with, and we proved that there are up to
. Which areܼయ, ܼమ ݔ ܼ, ܼ ݔ ܼ ݔ
. This covers for groups of order 8… Sylow’s theorem was
, where p and q are two distinct primes.
And there is only one group of such order up to isomorphism, which is ܼ
. This covers groups of order 15… Sylow’s theorem was also used
and there are only two Abelian groups of
ܼ. This covers order 12. Finally
groups of order one are the trivial groups. And all groups of order 1 are
abelian because the trivial subgroup of any group is a normal subgroup of
Abelian, cyclic, isomorphism, order, prime.
2010 Mathematics Subject Classification: 20F34, 20E40, 20D20, 20E36
How to cite this paper
Somkene Samuel "A Classification of
Groups of Small Order upto
Isomorphism" Published in
International
Journal of Trend in
Scientific Re
and Development
(ijtsrd), ISSN: 2456
6470, Volume
Issue-4, June 2020,
pp.627-631, URL:
www.ijtsrd.com/papers/ijtsrd31139.pdf
Copyright © 20
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
(http://creativecommons.org/licenses/
by/4.0)
The knowledge of Lagrange theorem and Sylow’s theorem
are important tools in the classification of groups. The
sylow’s first theorem helps us to present a group order in
t divide p. Lagrange
theorem helps us to know the possible divisors of a group.
Again the knowledge of the centre of a group, normal
subgroups contained in a group as well as direct product
of groups helps us to classify whether a group is abelian or
belian. Using the notation of Gorenstein [5], any
finite Abelian group G is Isomorphic to a direct product of
power order. Berkovich [1], added
will have the same number
ther. For example; Z6 ≅ Z2 ×
. The knowledge of normal subgroup is an
indispensable tool in the study of group classification. It
abelian groups. A
normal subgroup is a subgroup that is invariant under
conjugation by members of the group of which it is a part.
of the group ܩ is normal in ܩ
ܽ݊݀ ݊ ∈ ܰ. Written
Galois was the first to realize the
significance of normal subgroups. Dummit [11], normal
subgroups are imperative because they (and only they)
can be used to create quotient groups of the given group.
Fraleigh[12], the normal subgroup of G are specifically the
kernels of group homomorphisms with domain G, which
implies that they can be used to internally classify those
The study of the Centre of a group will equally help us to
know which groups are Abelian and those that are non
Abelian.
The Centre of a group
Let G be any group. The Centre of G is denoted by Z (G) =
{x ∈ G | xg = gx ∀ g ∈ G}
Thus the Centre of G consists of all those elements of G
which commute with every element of G. Note: if all
element of a group commutes with e
the Centre of the group is the group itself, we say that the
group is Abelian. Nevertheless, there are some groups that
its Centre is not the group itself. Those groups are called
non-Abelian groups. Roman, S. (2019 unpublished
dissertation), noted that all finite Abelian groups are built
from cyclic groups of prime
product. For symmetric group {Sn} of n
Abelian group. It is also important to note that all cyclic
groups denoted by Z are Abelian.
our knowledge of center of a group. The Centre of cyclic
groups gives us the group itself, which implies that all
cyclic groups are Abelian.
1. ࡳ࢛࢙࢘ ࢌ ࡻ࢘ࢊࢋ࢘ ࡼ࢘ࢋ
ࡼ࢙࢚࢘ . : ܷݎ݉ݏ݅ ݐ
.݁݉݅ݎ ݎ݁݀ݎ ݂ ݑݎ݃ ݁݊
International Journal of Trend in Scientific Research and Development (IJTSRD)
ISSN: 2456 – 6470
2020 Page 627
pto Isomorphism
Nsugbe, Nigeria
How to cite this paper: Ezenwobodo
Somkene Samuel "A Classification of
Groups of Small Order upto
Isomorphism" Published in
International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456-
6470, Volume-4 |
4, June 2020,
631, URL:
www.ijtsrd.com/papers/ijtsrd31139.pdf
Copyright © 2020 by author(s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
(http://creativecommons.org/licenses/
The study of the Centre of a group will equally help us to
know which groups are Abelian and those that are non –
Let G be any group. The Centre of G is denoted by Z (G) =
Thus the Centre of G consists of all those elements of G
which commute with every element of G. Note: if all
element of a group commutes with each other i.e. {Z(G)=G}
the Centre of the group is the group itself, we say that the
group is Abelian. Nevertheless, there are some groups that
its Centre is not the group itself. Those groups are called
Abelian groups. Roman, S. (2019 unpublished
ertation), noted that all finite Abelian groups are built
of prime-power order using direct
product. For symmetric group {Sn} of n ≥ 3 is not an
It is also important to note that all cyclic
groups denoted by Z are Abelian. This is from
our knowledge of center of a group. The Centre of cyclic
groups gives us the group itself, which implies that all
ࡼ࢘ࢋሺሻ ࢇࢊ
݅ݕ݈݊ ݏ݅ ݁ݎ݄݁ݐ ݉ݏ݄݅ݎ݉ݏ
IJTSRD31139](https://image.slidesharecdn.com/120aclassificationofgroupsofsmallorderuptoisomorphism-200716125155/85/A-Classification-of-Groups-of-Small-Order-upto-Isomorphism-1-320.jpg)
![International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD31139 | Volume – 4 | Issue – 4 | May-June 2020 Page 630
There are elements of G of order not equal to 1, p or q. any
such element must have order pq.
We can assume an element x of order p and y of order q: y
is a complement of x
|<x>| ≤ G and |<y>| ≤ G
F: (x, y) ⟶ x × y
Let F be a homomorphism from <x> × <y> to G,
we have the right to do that since <x> n <y> = {e}. By
Lagrange theorem, the divisors of prime (p) are {1 and p},
hence it follows that |x| and |y| have the same cardinality.
It suffices that F is an Isomorphism
<x> ≅<y> ≅ Zpq
G ≅<x> × <y> ≅Zp × Zq
this covers groups of order 15 …
Corollary 3.2: Every group of order Zpq is Isomorphic to
Zp × Zq and there is only one group of order pq.
ࡼ࢙࢚࢘ .
For every Abelian group of order p2q;
(i) G ≅ZP
2 × Zq
(ii) G ≅ ZP × Zp× Zq
Proof:
Suppose G is a finite group of order p2q
for all p, q distinct primes: p2 is not congruent to 1 mod p
and q is not congruent to1 mod p
then G is Abelian.
By sylow’s theorem
np = 1 + kp and it must divide p2q.
So, 1+ kp / q and because q is not congruent to 1 mod p
⟹ np = 1. This means we have a unique sylow p (G) for an
example of p in the group and is normal and also
Isomorphic to ZP
2 or ZP × ZP
Since q does not divide ଶ
− 1, therefore
nq = 1 + kq is not congruent to p, p2 . So we also have a
normal sylow q (G). Hence G is Abelian
G≅ ZP
2 × Zq
G≅ ZP × ZP × Zq
Another simpler way to see this is;
np
2
/ q = 1 mod p2 = {1, p, p2}
nq / p2 = 1 mod q = {1, q} = 1
Hence sylow q (G) is characteristically normal in G
i.e. we have a unique sylow q (G)
Let order p2q have x2, y. ∀ ܩ = p2q, and let x2 ∈ p2 and
y ∈ ݍ
| < ݔଶ
> | ≤ | ݏ݈ܽ ܩ < ݕ > | ≤ ܩ
Suppose ϴ: x2, y → x2 × y is a homomorphism that maps x2,
y to G.
Since y has order prime (q), and p and q are distinct;
x2 ∩ y = {e} and | x2| = | y |, hence x2 ≅ y ≅ x × x × y
G ≅ x2 × y
G≅ x × x × y
Hence; G≅Zp
2 x Zq
G ≅Zp x Zp x Zq
This covers abelian groups of order 12.
࢘ࢇ࢘࢟ . :
ܶℎ݁ ݎ݁݀ݎ ݂ ݏݑݎ݃ ݈ܾ݊ܽ݅݁ܽ ݓݐ ݕ݈݊ ݁ݎܽ ݁ݎଶ
,ݍ ݐݑ
݅ݎ݉ݏℎ݅ .݉ݏ
ࡾࢋࢇ࢘ . : There are (up to Isomorphism) exactly
three distinct non-abelian groups of order 12: the dihedral
group D6, the alternating groups A4, and a group T
generated by elements a and b such that | a | = 6, b2 = a3
and ba = a-1b.
Griess [4], the group T of order 12 is an example of a
dicyclic group. A presentation of the nth dicyclic group,
denoted Dicn , is again by (x | y) where x = { a, b} and Y =
{a2n, anb-2, b-1aba}/that is Dicn is generated by a and b,
where a and b satisfy the relations a2n = e, an = b2, and b-
1ab = a-1. The group Dicn is of order 4n. So the group T is
actually the third dicyclic group, Dic3. Gorenstein [5],
noted that the first dicyclic group is Isomorphic to Z4; for n
greater than or equal to 2, Dicn is non-Abelian. The second
dicyclic group is Isomorphic to the quaternions, Q8 ≅
Dic2. When n is a power of 2, Dicn is Isomorphic to a
“generated quaternion group”
4. CONCLUSION
Berkovich and Janko [2], any finite Abelian group G is
Isomorphic to a direct product of cyclic groups of prime –
power order. Moreover, this decomposition for G has the
same number of non-trivial factors of each other
For example;
Z6 ≅ Z2 × Z3
Z12 ≅Z3 × Z4
The study of the Centre of a group and normal subgroups
will equally help us to know which groups are Abelian and
those that are non – Abelian.
Note: “there is no known formula giving the number of
distinct {i.e. non Isomorphic} groups of order n, for every
n. however, we have the equipment to classify all groups of
order less than or equal to 15. For prime orders 2, 3, 5, 7,
11 and 13, there is only one group to each of these orders.
For orders 6, 10, 14 there are two non-Isomorphic groups
of order 4. Z4 and Z2 × Z2. There are five groups of order 8,
Z8, Z4 x Z2, Z2 × Z2 x Z2, Q8 and D4. There are two groups of
order 9 as Z9 and Z3 × Z3.
There are five groups of order 12, Z12, Z6 × Z2, A4, D6 and T.
And there is only one group of order 15, Z15.
All cyclic groups are Abelian. This is from our knowledge
of Centre of a group. The Centre of cyclic groups gives us
the group itself, which implies that all cyclic groups are
Abelian.
All finite Abelian groups are built from cyclic groups of
prime-power order using direct product. For symmetric
group {Sn} of n ≥ 3 is not an Abelian group.
We finish this paper with a table given the known groups
of order up to 15.](https://image.slidesharecdn.com/120aclassificationofgroupsofsmallorderuptoisomorphism-200716125155/85/A-Classification-of-Groups-of-Small-Order-upto-Isomorphism-4-320.jpg)
![International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD31139 | Volume – 4 | Issue – 4 | May-June 2020 Page 631
ORDER GROUP COMMENTS
1 Z1 The Trivial Group
2 Z2
3 Z3 ≅ A3
4
Z4
Klein4 – group V
≅ Z2 × Z2
The Smallest non-
cyclic group
5 Z5
6 Z6 ≅ Z2 × Z3
S3 ≅ D3
The Smallest non –
abelian group
7 Z7
8
Z8
Z2 × Z4
Z2 × Z2 × Z2
D4
Quaternion Q8
Non- Abelian
Non- Abelian
9 Z9
Z3 × Z3
10 Z10 ≅ Z2 × Z5
D5
Non – Abelian
11 Z11
12
Z12 ≅ Z3 × Z4
Z2 × Z6 ≅ Z2 × Z2
× Z3
D6 ≅ Z2 × D3
A4
Dic3 ≅ T
Non – Abelian
Non Abelian; smallest
group which shows
converse of Lagrange
theorem doesn’t hold
Non- Abelian, dicyclic
group of order 12
13 Z13
14 Z14 ≅ Z2 × Z7
D7
Non – Abelian
15 Z15 ≅ Z3 × Z5
There are 28 groups of order 15 or less, 20 of which are
Abelian.
REFERENCES
[1] Berkovich Y. Groups of prime order. Walter de
Gruyter, Berlin 2008.
[2] Berkovich Y and Z. Janko. Groups of prime order (vol
11). Walter de Gruyter, Berlin, 2008.
[3] Boya L.J and Rivera C. groupos abelianos finitos.
Mirada categorial. Gazeta de la RSME 13:229- 244,
2010.
[4] Griess R. L. Twelve sporadic groups, Springer, Berlin.
1998.
[5] Gorenstein D. Finite groups. New York 2000.
[6] Sautoy M. Du. Symmetry. Harper-Collins, 2008
[7] Huppert B. Endliche Gruppen I. springer, 1997.
[8] Cox David. Galois Theorem. Wiley-interscience.
Hoboken, NJ,xx+559pp.MR2119052, 2004
[9] Jacobson, Nathan (2009), basic algebra. 1 (2nd ed.).
Dover. ISBN 978-0-486-47189-1.
[10] The GAP Group, GAP-Groups, algorithms and
programming, version 4.8.4 (2016);
http://www.gap-system.org.
[11] Dummit, David S.; Foote, Richard M. (2004). Abstract
Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-
43334-9.
[12] Fraleigh, John B. (2003). A First Course in Abstract
Algebra (7th ed.). Addison-Wesley. ISBN 978-0-321-
15608-2](https://image.slidesharecdn.com/120aclassificationofgroupsofsmallorderuptoisomorphism-200716125155/85/A-Classification-of-Groups-of-Small-Order-upto-Isomorphism-5-320.jpg)
A Classification of Groups of Small Order upto Isomorphism
- 1. International Journal of Trend in Scientific Research and Development (IJTSRD) Volume 4 Issue 4, June 2020 @ IJTSRD | Unique Paper ID – IJTSRD31139 A Classification of Groups Ezenwobodo Somkene Samuel Department of Mathematics, Nwafor Orizu College ABSTRACT Here we classified groups of order less than or equal to 15. We proved that there is only one group of order prime up to isomorphism, and that all groups of order prime (P) are abelian groups. This covers groups of order 2,3,5,7,11,13….Again we were able to prove that there are up to isomorphism only two groups of order 2p, where p is prime and p this is ܼଶ ≅ ܼଶ x ܼ. (Where Z represents cyclic group), and dihedral group of the p-gon). This covers groups of order 6, 10, 14….. And we proved that up to isomorphism there are only two groups of order P And these are ܼమ and ܼ ݔ ܼ. This covers groups of order 4, 9…..Groups of order P3 was also dealt with, and we proved that there are up to isomorphism five groups of order P3. Whi ܼ, ܦయ and ܳయ. This covers for groups of order 8… Sylow’s theorem was used to classify groups of order ݍ, where p and q are two distinct primes. And there is only one group of such order up to isomor ≅ ܼ ݔ ܼ. This covers groups of order 15… Sylow’s theorem was also used to classify groups of order ଶ ݍ and there are only two Abelian groups of such order which are Zp 2 q and ܼ ݔ ܼ ݔ groups of order one are the trivial groups. And all groups of order 1 are abelian because the trivial subgroup of any group is a normal subgroup of that group. KEYWORDS: Abelian, cyclic, isomorphism, order, prime. 2010 Mathematics Subject Classification: INTRODUCTION The knowledge of Lagrange theorem and Sylow’s theorem are important tools in the classification of groups. The sylow’s first theorem helps us to present a group order in the form of ݍ where q doesn’t divide p. Lagrange theorem helps us to know the possible divisors of a group. Again the knowledge of the centre of a group, normal subgroups contained in a group as well as direct product of groups helps us to classify whether a group is abelian or non-abelian. Using the notation of Gorenstein [5], any finite Abelian group G is Isomorphic to a direct product of cyclic groups of prime–power order. Berkovich [1], added that this decomposition for ܩ will have the same number of non-trivial factors of each other. For example; Z Z3, Z12 ≅ Z3 × Z4. The knowledge of normal subgroup is an indispensable tool in the study of group classification. It will aid to differentiate abelian and non-abelian groups. A normal subgroup is a subgroup that is invariant conjugation by members of the group of which it is a part. In other words, a subgroup ܰ of the group if and only if ݃݊݃ିଵ ∈ ܰ for all ݃ ∈ ݀݊ܽ ܩ as ܰ ⊲ .ܩ Evariste Galois was the first to realize the significance of normal subgroups. Dummit [11], normal subgroups are imperative because they (and only they) can be used to create quotient groups of the given group. Fraleigh[12], the normal subgroup of G are specifical kernels of group homomorphisms with domain G, which implies that they can be used to internally classify those homomorphisms. International Journal of Trend in Scientific Research and Development (IJTSRD) 2020 Available Online: www.ijtsrd.com e-ISSN: 2456 31139 | Volume – 4 | Issue – 4 | May-June 2020 f Groups of Small Order upto Isomorphism Ezenwobodo Somkene Samuel f Mathematics, Nwafor Orizu College of Education, Nsugbe, Nigeria Here we classified groups of order less than or equal to 15. We proved that there is only one group of order prime up to isomorphism, and that all groups of order prime (P) are abelian groups. This covers groups of order o prove that there are up to isomorphism only two groups of order 2p, where p is prime and p≥3, and . (Where Z represents cyclic group), and ܦ (the gon). This covers groups of order 6, 10, 14….. And roved that up to isomorphism there are only two groups of order P2. . This covers groups of order 4, 9…..Groups was also dealt with, and we proved that there are up to . Which areܼయ, ܼమ ݔ ܼ, ܼ ݔ ܼ ݔ . This covers for groups of order 8… Sylow’s theorem was , where p and q are two distinct primes. And there is only one group of such order up to isomorphism, which is ܼ . This covers groups of order 15… Sylow’s theorem was also used and there are only two Abelian groups of ܼ. This covers order 12. Finally groups of order one are the trivial groups. And all groups of order 1 are abelian because the trivial subgroup of any group is a normal subgroup of Abelian, cyclic, isomorphism, order, prime. 2010 Mathematics Subject Classification: 20F34, 20E40, 20D20, 20E36 How to cite this paper Somkene Samuel "A Classification of Groups of Small Order upto Isomorphism" Published in International Journal of Trend in Scientific Re and Development (ijtsrd), ISSN: 2456 6470, Volume Issue-4, June 2020, pp.627-631, URL: www.ijtsrd.com/papers/ijtsrd31139.pdf Copyright © 20 International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/ by/4.0) The knowledge of Lagrange theorem and Sylow’s theorem are important tools in the classification of groups. The sylow’s first theorem helps us to present a group order in t divide p. Lagrange theorem helps us to know the possible divisors of a group. Again the knowledge of the centre of a group, normal subgroups contained in a group as well as direct product of groups helps us to classify whether a group is abelian or belian. Using the notation of Gorenstein [5], any finite Abelian group G is Isomorphic to a direct product of power order. Berkovich [1], added will have the same number ther. For example; Z6 ≅ Z2 × . The knowledge of normal subgroup is an indispensable tool in the study of group classification. It abelian groups. A normal subgroup is a subgroup that is invariant under conjugation by members of the group of which it is a part. of the group ܩ is normal in ܩ ܽ݊݀ ݊ ∈ ܰ. Written Galois was the first to realize the significance of normal subgroups. Dummit [11], normal subgroups are imperative because they (and only they) can be used to create quotient groups of the given group. Fraleigh[12], the normal subgroup of G are specifically the kernels of group homomorphisms with domain G, which implies that they can be used to internally classify those The study of the Centre of a group will equally help us to know which groups are Abelian and those that are non Abelian. The Centre of a group Let G be any group. The Centre of G is denoted by Z (G) = {x ∈ G | xg = gx ∀ g ∈ G} Thus the Centre of G consists of all those elements of G which commute with every element of G. Note: if all element of a group commutes with e the Centre of the group is the group itself, we say that the group is Abelian. Nevertheless, there are some groups that its Centre is not the group itself. Those groups are called non-Abelian groups. Roman, S. (2019 unpublished dissertation), noted that all finite Abelian groups are built from cyclic groups of prime product. For symmetric group {Sn} of n Abelian group. It is also important to note that all cyclic groups denoted by Z are Abelian. our knowledge of center of a group. The Centre of cyclic groups gives us the group itself, which implies that all cyclic groups are Abelian. 1. ࡳ࢛࢙࢘ ࢌ ࡻ࢘ࢊࢋ࢘ ࡼ࢘ࢋ ࡼ࢙࢚࢘ . : ܷݎ݉ݏ݅ ݐ .݁݉݅ݎ ݎ݁݀ݎ ݂ ݑݎ݃ ݁݊ International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 – 6470 2020 Page 627 pto Isomorphism Nsugbe, Nigeria How to cite this paper: Ezenwobodo Somkene Samuel "A Classification of Groups of Small Order upto Isomorphism" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456- 6470, Volume-4 | 4, June 2020, 631, URL: www.ijtsrd.com/papers/ijtsrd31139.pdf Copyright © 2020 by author(s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/ The study of the Centre of a group will equally help us to know which groups are Abelian and those that are non – Let G be any group. The Centre of G is denoted by Z (G) = Thus the Centre of G consists of all those elements of G which commute with every element of G. Note: if all element of a group commutes with each other i.e. {Z(G)=G} the Centre of the group is the group itself, we say that the group is Abelian. Nevertheless, there are some groups that its Centre is not the group itself. Those groups are called Abelian groups. Roman, S. (2019 unpublished ertation), noted that all finite Abelian groups are built of prime-power order using direct product. For symmetric group {Sn} of n ≥ 3 is not an It is also important to note that all cyclic groups denoted by Z are Abelian. This is from our knowledge of center of a group. The Centre of cyclic groups gives us the group itself, which implies that all ࡼ࢘ࢋሺሻ ࢇࢊ ݅ݕ݈݊ ݏ݅ ݁ݎ݄݁ݐ ݉ݏ݄݅ݎ݉ݏ IJTSRD31139
- 2. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31139 | Volume – 4 | Issue – 4 | May-June 2020 Page 628 ࡸࢋࢇ . : ݈ܿ݅ܿݕܿ ܽ ݏ݅ ݁݉݅ݎܲ ݎ݁݀ݎܱ ݂ ݑݎܩ ݕݎ݁ݒܧ ݃,ݑݎ ℎ݁݊ܿ݁ ℎܽ݁݊ ݕ݈݊ ݏ ݃݁݊݁.݅ ݎݐܽݎ ݁. ݅ .݂݈݁ݏݐ ࡼ࢘ࢌ: Let g ∈ G be arbitrary chosen by Lagrange theorem Let G be a finite group and H a subgroup of G; H < G ܶℎ݁݊ //ܩ/ ݏ݁݀݅ݒ݅݀ /ܪ |g| = |<g>| will divide |G|=P hence |g| = g1 = e=1 i.e. g is identity element or |g| = p g1 × g2 ×…..× gp-1 <g> = {g.g1... gp-1. e}= G ⇒ G is cyclic By G being cyclic, it means it has only one element generator i.e. it has only one element that generates all the other elements of G. Therefore, it’s isomorphic to Additive group of integers Modp. G≅Zp From the proof, we can see that there is only one divisor of G if G is prime. And that is G itself since 1 is the identity. Hence G ≅Zp Iff G is cyclic. The consequence of this is that groups of order 1, 2, 3, 5, 7, 11, 13… have only one group Up to Isomorphism. Corollary 1.3: There is only one group of order Prime (P) up to isomorphism and it’s Abelian. ࡼ࢙࢚࢘ . : ,2 ݎ݁݀ݎ ݂ ݑݎ݃ ܽ ݏ݅ ܩ ݁ݏݑݏ ݓℎ݁݁ݎ ≥ 3, ݅ .݁݉݅ݎ ܽ ݏ ݐ݅ܧℎ݁:ݎ A. ≅ ܼଶ ݅ ݎ ݑݎ݃ ݈ܿ݅ܿݕܿ ܽ ݏ B. ܩ ≅ ܦ ݅ݎ݉ݏܫ ݏℎ݅ܿ ݐ ݐℎ݁ ݀݅ℎ݁݀ݐ ݂ ݑݎ݃ ݈ܽݎℎ݁ ܲ − ݃ .݊ ࡼ࢘ࢌ: By disjunctive syllogism i.e. either G≅Z2p or G≅Dp suppose G is not cyclic According to Lagrange theorem Let G be a finite group and H < G be a subgroup of G. /./ܩ/ ݏ݁݀݅ݒ݅݀ /ܪ But the only divisors of 2 are 1, 2, ܲ, 2ܲ. But since we assume G is not cyclic then no order 2p can exist. Because if it does exist then it can generate all the other elements of the group so we are left with; 1, 2, ܲ. ( 1 = ݅݀݁݊݁( ݐ݈݁ ݕݐ݅ݐሻሻ Suppose there is no order of P in the Group i.e. by contradiction then: All elements of the group would be order 1 or 2 Then we would have an Evolution i.e every non-identity element will be its own inverse ⇒ that G = Abelian. Let the two elements of G be a & b G = {e, a, b, ab} Since the elements of G is closed and contains an identity element and is finite; Hence {e, a, b, ab} < G i.e. {e, a, b, ab} is a subgroup of G But it’s a subgroup that has four elements. But G cannot have four elements because 4 does not divide 2p by Lagrange theorem; surely we don’t know what 2p is but we know that 2 goes into P once because 2 does not divide P if 4 does not divide 2P, hence /a/ = P; that means the first contradiction cannot hold. If /a / = p then <a> (generator) ⇒ ܽ, ܽଶ , ܽଷ … ܽ ିଵ Let’s take another arbitrary element b If b is not an element of <a>, we can claim that / b/ = 2 If not / b / = P and we don’t want it; because by product theorem < ܽ > ݊ < ܾ > = ሼ݁ሽ/= /ܭܪ ̸ = /݇ ݊ ܪ / × /݇ ܪ ܲଶ ≥ 2 i.e. P2 1 is not contained in 2P, iif H and K are both primes. Hence a contradiction, So |b| ≠ P ܶℎ= / ܾ/ ݏݑ 2 Finally ab = (ab)-1 = b-1 a-1 = ba-1 for all of b an evolution ab = ba-1 and from the dihedral group we have that; ܦଶ = < ܽ, ܾ / ܽ = ݕଶ = 1; ܾܽ = ܾܽିଵ > The same way a, b relate in the cyclic group of order 2P, is the same way a, b relate in the dihedral group of the p-gon ⇒ G ⇒ dihedral and G ⇒ cyclic the consequences of this proof is that a group of order 2p is either isomorphic to Z2p or Isomorphic to Dp (i.e. the dihedral group of the p-gon) The result of this proof implies that there are only two group of order 6, 10, 14… 2.2.2.2. ࡳ࢛࢙࢘ ࢌ ࡼ࢘ࢋ ࡿ࢛ࢇ࢘ࢋ ( ሻ ࢇࢊ ࢘ࢋ ࢉ࢛࢈ࢋ ( ሻ ࢀࢎࢋ࢘ࢋ . There are only two groups of order P2 ࢋࢇ . Let P be prime: there are only two groups Upto Isomorphism of order P2 Proof: suppose G is a P2 group. It is Abelian. According to Lagrange Theorem the divisors of P2 are 1, P, P2 Let x: x has order P2 then G= <x> generates all the elements of the group G = P2. So G is cyclic. This satisfies the earlier notation that G is Abelian. G≅Zp2 Now assume that there is no element of order P2. This means that every element which is not the identity has order P. pick x order P. since <x> ≤G, you can take another order P element y in the complement of <x>.
- 3. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31139 | Volume – 4 | Issue – 4 | May-June 2020 Page 629 Now ϴ: (u, v) → uv yields a homomorphism from <x> × <y> to G. Note that <x> n <y> = <e>, so the latter is injective. Since by Lagrange theorem both groups have same cardinality, it follows that ϴ is an Isomorphism. If <y> is a complement of <x> it suffices that only the identity element will be the intersection since they are different primes. And of course we all know that the cardinality of primes is always the same. It implies that ϴ is an Isomorphism. Finally since <x> ≅ < y> ≅ Zp G≅ <x> x <y> ≅ Zp x Zp So G is either Isomorphic to Zp 2 or to Zp × Zp of course the implication of this is that every group of P2 is either Zp2 or Zp×Zp i.e. there are only two groups of order P2 up to Isomorphism. This covers groups of order 4, 9… ࡼ࢙࢚࢘ . There are five groups of order P3 either 1. G ≅ Zp3 ≅Zp × Zp2 ≅Zp × Zp × Zp Or 2. G ≅Dp3 ≅Qp 3 Proof: From the proposition above (2.3) we can deduce that by transitive property that G ≅Zp3 G ≅ ZP × Zp2 G ≅ZP × ZP ×ZP G ≅ DP 3 G ≅QP 3 That’s five groups in total by disjunctive syllogism i.e. either / or, Suppose G is not cyclic by Lagrange theorem Let G be a finite group and H<G be a subgroup of G but the only divisors of P3 are 1, P, P2 , P3 But we can’t take order P3 because <x> = p3 will generate all the members of the group making it cyclic. So we have 1, P, P2 Suppose we take |b| = p; <b> will generate all the members of P and suppose we take |a|= P2; <a> will generate all the members of P2. Hence group of order P3 must contain some cyclic groups. But let order P3 have x, y & z; recall G = Zp 3 hence |<x>| ≤ G and |<y>| ≤ G also |<z>| ≤ G F: (x, y, z) ⟶ x × y × z Let F be a homomorphism that map <x> × <y> × < z > to G Of course since {x, y, z} ∈ P2 and also {x, y, z} is contained in P2 Then <x> n <y> n <z> = e and they must have the same cardinality Iff {x, y, z} are subgroups of order P2 and are contained in G i.e. G = P3 then G ≅ <x> ≅ <y> ≅ <z> ≅ ZP 3 G≅ <x> × <y> x < z> ≅ ZP 2 x Zp G ≅ <x> × <y> × <z> ≅ ZP× ZP × ZP But if |b| = p & | a| = P2; then b = |2| Hence b is an evolution; therefore being its own inverse ab = (ab) – 1 = b-1a-1 But b is an evolution ⇒ ab = ba-1 and from the dihedral group we know that D2n = <a, b / an = y2 = 1; ab = ba-1> And also of the Quaternion group Q4n = < a, b / a2n = y4 = 1; ab = ba-1> the same way a, b relate to the cyclic group of order p3, is the same way a, b relate to the dihedral and Quaternion groups of same order. G ⇒ dihedral G ⇒ Quartenion & G ⇒ cyclic This covers groups of order 8, 27… 3. ࡳ࢛࢙࢘ ࢌ ࢘ࢊࢋ࢘ ࢇࢊ ࡼ࢙࢚࢘ . If G is a group of order pq for some primes, pq such that p>q and q doesn’t divide (p – 1) then G ≅ Zpq≅ Zp x Zq Proof: we can find a unique sylow p and sylow q subgroups of G. By the third sylow theorem Let Sq be sylow q & Sp be sylow p Sp | q and Sp = 1 + kp Since q is a prime the first condition gives Sp = 1 or Sp= q Since p>q the second condition implies then that SP = 1 similarly let Sq be the number of sylow q – subgroups of G We have Sq | p and Sq = 1 + kq the first condition gives Sq = 1 or Sq = P. If Sq = P then the first second condition gives P = 1 + Kq, or P- 1 = Kq this is however impossible since q doesn’t divide (p – 1). Therefore, we have Sq = 1 Another way to see this is: Sp | q & Sp ≡ 1 modp ⟹ {1, kp + 1} ∀ k ∈ ℤ Sq | p & Sq ≡ 1 modq ⟹ {1, Kq +1} ∀ k ∈ ℤ Sq | P = {1} or {Sp n Sq} = 1 hence since Sq | P =1, Sq = 1 & Sp = 1 It means we have a unique sylow P subgroup and a unique sylow q subgroup. By the second law of Sylow’s theorem. Every element of G of order P belongs to the subgroup P and every element of order q belongs to the subgroup Q. It follows that G contains exactly P-1 elements of order P. exactly q-1 elements of order q and one trivial element of order 1. Since for p, q we have pq > (p – 1) + (q – 1) + 1
- 4. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31139 | Volume – 4 | Issue – 4 | May-June 2020 Page 630 There are elements of G of order not equal to 1, p or q. any such element must have order pq. We can assume an element x of order p and y of order q: y is a complement of x |<x>| ≤ G and |<y>| ≤ G F: (x, y) ⟶ x × y Let F be a homomorphism from <x> × <y> to G, we have the right to do that since <x> n <y> = {e}. By Lagrange theorem, the divisors of prime (p) are {1 and p}, hence it follows that |x| and |y| have the same cardinality. It suffices that F is an Isomorphism <x> ≅<y> ≅ Zpq G ≅<x> × <y> ≅Zp × Zq this covers groups of order 15 … Corollary 3.2: Every group of order Zpq is Isomorphic to Zp × Zq and there is only one group of order pq. ࡼ࢙࢚࢘ . For every Abelian group of order p2q; (i) G ≅ZP 2 × Zq (ii) G ≅ ZP × Zp× Zq Proof: Suppose G is a finite group of order p2q for all p, q distinct primes: p2 is not congruent to 1 mod p and q is not congruent to1 mod p then G is Abelian. By sylow’s theorem np = 1 + kp and it must divide p2q. So, 1+ kp / q and because q is not congruent to 1 mod p ⟹ np = 1. This means we have a unique sylow p (G) for an example of p in the group and is normal and also Isomorphic to ZP 2 or ZP × ZP Since q does not divide ଶ − 1, therefore nq = 1 + kq is not congruent to p, p2 . So we also have a normal sylow q (G). Hence G is Abelian G≅ ZP 2 × Zq G≅ ZP × ZP × Zq Another simpler way to see this is; np 2 / q = 1 mod p2 = {1, p, p2} nq / p2 = 1 mod q = {1, q} = 1 Hence sylow q (G) is characteristically normal in G i.e. we have a unique sylow q (G) Let order p2q have x2, y. ∀ ܩ = p2q, and let x2 ∈ p2 and y ∈ ݍ | < ݔଶ > | ≤ | ݏ݈ܽ ܩ < ݕ > | ≤ ܩ Suppose ϴ: x2, y → x2 × y is a homomorphism that maps x2, y to G. Since y has order prime (q), and p and q are distinct; x2 ∩ y = {e} and | x2| = | y |, hence x2 ≅ y ≅ x × x × y G ≅ x2 × y G≅ x × x × y Hence; G≅Zp 2 x Zq G ≅Zp x Zp x Zq This covers abelian groups of order 12. ࢘ࢇ࢘࢟ . : ܶℎ݁ ݎ݁݀ݎ ݂ ݏݑݎ݃ ݈ܾ݊ܽ݅݁ܽ ݓݐ ݕ݈݊ ݁ݎܽ ݁ݎଶ ,ݍ ݐݑ ݅ݎ݉ݏℎ݅ .݉ݏ ࡾࢋࢇ࢘ . : There are (up to Isomorphism) exactly three distinct non-abelian groups of order 12: the dihedral group D6, the alternating groups A4, and a group T generated by elements a and b such that | a | = 6, b2 = a3 and ba = a-1b. Griess [4], the group T of order 12 is an example of a dicyclic group. A presentation of the nth dicyclic group, denoted Dicn , is again by (x | y) where x = { a, b} and Y = {a2n, anb-2, b-1aba}/that is Dicn is generated by a and b, where a and b satisfy the relations a2n = e, an = b2, and b- 1ab = a-1. The group Dicn is of order 4n. So the group T is actually the third dicyclic group, Dic3. Gorenstein [5], noted that the first dicyclic group is Isomorphic to Z4; for n greater than or equal to 2, Dicn is non-Abelian. The second dicyclic group is Isomorphic to the quaternions, Q8 ≅ Dic2. When n is a power of 2, Dicn is Isomorphic to a “generated quaternion group” 4. CONCLUSION Berkovich and Janko [2], any finite Abelian group G is Isomorphic to a direct product of cyclic groups of prime – power order. Moreover, this decomposition for G has the same number of non-trivial factors of each other For example; Z6 ≅ Z2 × Z3 Z12 ≅Z3 × Z4 The study of the Centre of a group and normal subgroups will equally help us to know which groups are Abelian and those that are non – Abelian. Note: “there is no known formula giving the number of distinct {i.e. non Isomorphic} groups of order n, for every n. however, we have the equipment to classify all groups of order less than or equal to 15. For prime orders 2, 3, 5, 7, 11 and 13, there is only one group to each of these orders. For orders 6, 10, 14 there are two non-Isomorphic groups of order 4. Z4 and Z2 × Z2. There are five groups of order 8, Z8, Z4 x Z2, Z2 × Z2 x Z2, Q8 and D4. There are two groups of order 9 as Z9 and Z3 × Z3. There are five groups of order 12, Z12, Z6 × Z2, A4, D6 and T. And there is only one group of order 15, Z15. All cyclic groups are Abelian. This is from our knowledge of Centre of a group. The Centre of cyclic groups gives us the group itself, which implies that all cyclic groups are Abelian. All finite Abelian groups are built from cyclic groups of prime-power order using direct product. For symmetric group {Sn} of n ≥ 3 is not an Abelian group. We finish this paper with a table given the known groups of order up to 15.
- 5. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31139 | Volume – 4 | Issue – 4 | May-June 2020 Page 631 ORDER GROUP COMMENTS 1 Z1 The Trivial Group 2 Z2 3 Z3 ≅ A3 4 Z4 Klein4 – group V ≅ Z2 × Z2 The Smallest non- cyclic group 5 Z5 6 Z6 ≅ Z2 × Z3 S3 ≅ D3 The Smallest non – abelian group 7 Z7 8 Z8 Z2 × Z4 Z2 × Z2 × Z2 D4 Quaternion Q8 Non- Abelian Non- Abelian 9 Z9 Z3 × Z3 10 Z10 ≅ Z2 × Z5 D5 Non – Abelian 11 Z11 12 Z12 ≅ Z3 × Z4 Z2 × Z6 ≅ Z2 × Z2 × Z3 D6 ≅ Z2 × D3 A4 Dic3 ≅ T Non – Abelian Non Abelian; smallest group which shows converse of Lagrange theorem doesn’t hold Non- Abelian, dicyclic group of order 12 13 Z13 14 Z14 ≅ Z2 × Z7 D7 Non – Abelian 15 Z15 ≅ Z3 × Z5 There are 28 groups of order 15 or less, 20 of which are Abelian. REFERENCES [1] Berkovich Y. Groups of prime order. Walter de Gruyter, Berlin 2008. [2] Berkovich Y and Z. Janko. Groups of prime order (vol 11). Walter de Gruyter, Berlin, 2008. [3] Boya L.J and Rivera C. groupos abelianos finitos. Mirada categorial. Gazeta de la RSME 13:229- 244, 2010. [4] Griess R. L. Twelve sporadic groups, Springer, Berlin. 1998. [5] Gorenstein D. Finite groups. New York 2000. [6] Sautoy M. Du. Symmetry. Harper-Collins, 2008 [7] Huppert B. Endliche Gruppen I. springer, 1997. [8] Cox David. Galois Theorem. Wiley-interscience. Hoboken, NJ,xx+559pp.MR2119052, 2004 [9] Jacobson, Nathan (2009), basic algebra. 1 (2nd ed.). Dover. ISBN 978-0-486-47189-1. [10] The GAP Group, GAP-Groups, algorithms and programming, version 4.8.4 (2016); http://www.gap-system.org. [11] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471- 43334-9. [12] Fraleigh, John B. (2003). A First Course in Abstract Algebra (7th ed.). Addison-Wesley. ISBN 978-0-321- 15608-2