Classification of Groups and Homomorphism -By-Rajesh Bandari Yadav

Classification of Groups
and
Homomorphism
By
Rajesh Bandari Yadav

Content
Chapter-1Chapter-1 Chapter-2Chapter-2 ReferencesReferences
Classification
of Groups of
Order
Up to 8
Homomorphism
Of
Groups
References

1. 1-Order Group1. 1-Order Group
Chapter-1Chapter-1
((Classification of Groups of Order Up to 8)
IntroductionIntroduction

2.1.Introduction2.1.Introduction
2.2.Properties of Homomorphism2.2.Properties of Homomorphism
2.3.Homomorphisms Of Finite Cyclic Group To Finite Cyclic Groups2.3.Homomorphisms Of Finite Cyclic Group To Finite Cyclic Groups
2.4.Method2.4.Method
Chapter-2Chapter-2
(HOMOMORPHISM OF GROUPS(HOMOMORPHISM OF GROUPS)
2.5.One-to-one homomorphism2.5.One-to-one homomorphism
2.6.One-to-one homomorphism from Cyclic group to cyclic group2.6.One-to-one homomorphism from Cyclic group to cyclic group
2.7.On-to homomorphism2.7.On-to homomorphism
2.8.On-to homomorphism from Cyclic group to cyclic group2.8.On-to homomorphism from Cyclic group to cyclic group
2.9.How to GAP2.9.How to GAP

Introduction
About Groups :
In abstract algebra, a finite group is a mathematical group with a
finite number of elements. A group is a set of elements together with
an operation which associates, to each ordered pair of elements, an
element of the set. With a finite group, the set is finite.
During the twentieth century, mathematicians investigated some
aspects of the theory of finite groups in great depth, especially the
local theory of finite groups and the theory of solvable and nilpotent
groups. As a consequence, the complete classification of finite
simple groups was achieved, meaning that all those simple groups
from which all finite groups can be built are now known.
During the second half of the twentieth century, mathematicians
such as Chevalley and Steinberg also increased our understanding
of finite analogs of classical groups, and other related groups. One
such family of groups is the family of general linear groups over
finite fields.

Application Of Groups:
Finite groups often occur when considering symmetry of
mathematical or physical objects, when those objects admit just a
finite number of structure-preserving transformations. The theory of
Lie groups, which may be viewed as dealing with "continuous
symmetry", is strongly influenced by the associated Weyl groups.
These are finite groups generated by reflections which act on a
finite-dimensional Euclidean space. The properties of finite groups
can thus play a role in subjects such as theoretical physics and
chemistry

About GAP:
GAP (Groups, Algorithms and Programming) is a computer algebra
system for computational discrete algebra with particular emphasis
on computational group theory.
GAP was developed at Lehrstuhl D für Mathematik (LDFM), RWTH
Aachen, Germany from 1986 to 1997. After the retirement of J.
Neubüser from the chair of LDFM, the development and maintenance
of GAP was coordinated by the School of Mathematical and
Computational Sciences at the University of St Andrews, Scotland. In
the summer of 2005 coordination was transferred to an equal
partnership of four 'GAP Centres', located at the University of St
Andrews, RWTH Aachen, Technische Universität Braunschweig, and
Colorado State University at Fort Collins

:
G={e} is a group w.r.to * of order 1.
1-Order Group
* e
e e
Cayleys diagram
Properties:
Each element order is one
G≅ (Z1 ,+1)
Subgroups:
G does not have any proper subgroups
G have only one sub group that is trivial

2-Order Group:
G={e , a} is a group w.r.to *.
* a a
e e a
a a e
Cayley diagram: Properties:
 G≅(Z2 , +2)
 G is a cyclic group
 G is simple group
 G does not have any proper
subgroups

G={e ,a ,b} is a cyclic group w.r.to *.
* e a b
e e a b
a a b e
b b e a
3-Order group
G is isomorphic to (Z3 ,+3)
G is a cyclic group.
G is simple group.

4-Order Group
 Case-i: G={e ,a , b , c } is an abelian group w.r.to *
* e a b c
e e a b c
a a e c b
b b c e a
c c b a e
 Here ‘e’ is the identity element
 a * a=e; b*b=e ; c*c=e.
 ab=ba , ac=ca, bc=cb,
G is an abelian Group.
G={<a,b> ; a2
=b2
=e,ab=ba=c}
G≅Z2XZ2≅U(8)≅(P(X),∆) where X={1,2}
Simply every 4-Order Group(Non-
cyclic) is isomorphic Z2XZz
G is NOT simple group

4-Order Group
Order of elements:
O(e)=1
O(a)=O(b)=O(c)=2.
i.e each element order is 2
Subgroups:
G={e, a, b, c}
H={e, a} K={e, b} U={e , c}
{e}
 Every proper sub group is isomorphic to (Z2 ,+2)
Total numbers of subgroups are 5.
G is abelian group that implies Every proper sub
group is Normal sub group

 Case-ii):G={e , a , b , c} is a cyclic group w.r.to *.
* e a b c
e e a b c
a a e c b
b b c a e
c c b e a
 ‘e’ is the identity element
 a *a=e; b*c=c*b=e.
 ab=ba; ac=ca;bc=cb.
G is a cyclic group.
G=<b>=<c>
G≅(Z4 ,+4)
G is NOT simple group
4-Order Group

Order of elements:
O(e)=1;
O(a)=2;
O(b)=O(c)=4
Subgroups:
G={e , a , b , c}
H={e ,a}
{e}
 2-order sub group is the only one proper subgroup of G
which isomorphic to (Z2, +2).
 Total numbers of subgroups are 3.
 G is cyclic group that implies every subgroup of G is Normal
4-Order Group

5-Order group:
 G={e ,a , b ,c ,d } is Cyclic group w.r.to *.
* e a b c d
e e a b c d
a a b c d e
b b c d e a
c c d e a b
d d e a b c
Caylay diagram :
Properties:
G is Cyclic group
G is Isomorphic to (Z5 ,+5)
G does not have any proper subgroups.
G is simple group.

6-Order Group:
 Case-i): G={e ,a1 , a2 ,a3 , a4 ,a5} is a non-abelian group w.r.to *.
Caylay diagram :
Properties:
G is a non-abelian group
G ≅D3≅S3≅GL(Z2,2)
Z(G)={e}
G is not simple group.
G is the first even order non-abelian
group
* e a1 a2 a3 a4 a5
e e a1 a2 a3 a4 a5
a1 a1 e a3 a4 a5 a2
a2 a2 a4 e a5 a3 a1
a3 a3 a5 a4 e a1 a2
a4 a4 a2 a1 a3 a5 e
a5 a1 a3 a2 a4 a4 e

Order of elements:
O(e)=1
O(a1)=O(a2)=O(a3)=2
O(a4)=O(a5)=3
Subgroups & Normal subgroups:
G={e , a1 , a2 , a3 , a4 , a5}
H1={e , a4 , a5} H2={e ,a1} H3={e , a2} H4={e , a3}
{e}
 Every sub group of G is Cyclic subgroup.
 H2, H3,H4 are isomorphic to (Z2 ,+2)
 H1={e ,a4 , a5} is isomorphic to (Z3 ,+3)
 Total numbers of subgroups are 6.
6-Order Group

 Case-ii) :G={ e , a1 , a2, a3 , a4 , a5} is Cyclic group w.r.to *.
* e a1 a2 a3 a4 a5
e e a1 a2 a3 a4 a5
a1 a1 a2 a3 a4 a5 e
a2 a2 a3 a4 a5 e a1
a3 a3 a4 a5 e a1 a2
a4 a4 a5 e a1 a2 a3
a5 a5 e a1 a2 a3 a4
Caylay diagram:
Properties:
G is Cyclic group.
G is Isomorphic to (Z6 ,+6)
G is not simple group
6-Order Group

Subgroups:
G={e , a1 , a2 ,a3 , a4 , a5 }
H={e , a3} H={e ,a2 , a4}
H={e}
 Every subgroup of G is cyclic
 Their a unique subgroups of orders 2,3 of G .
 Total numbers of subgroups of G are 4.
6-Order Group

7-Order group:
 G={e ,a1 , a2, a3, a4 , a5 , a6 } is a cyclic group w.r.to *
* e a1 a2 a3 a4 a5 a6
e e a1 a2 a3 a4 a5 a6
a1 a1 a2 a3 a4 a5 a6 e
a2 a2 a3 a4 a5 a6 e a1
a3 a3 a4 a5 a6 e a1 a2
a4 a4 a5 a6 e a1 a2 a3
a5 a5 a6 e a1 a2 a3 a4
a6 a6 e a1 a2 a3 a4 a5
Caylay diagram:

Properties:
G is Cyclic Group.
G=<a1>=<a2>=<a3>=<a4>=<a5>=<a6>
G≅(Z7 ,+7)
G is a simple group.
Subgroups:
Total numbers of subgroups of G are only two
7-Order group:

8-Order Groups
 Q8={1 ,-1 , i, -i, j , -j , k , -k} is a non-abelian group w.r.to
multiplication
. 1 -1 i -i j -j k -k
1 1 -1 i -i j -j k -k
-1 -1 1 -i i -j j -k k
I i -i -1 1 k -k j -j
-i -i i 1 -1 -k K -j J
J j -j -k K -1 1 i -i
-j -j j k -k 1 -1 -i i
K k -k -j J -i I -1 1
-k -k k j -j i -i 1 -1
Caylay diagram:
 Properties:
 Q8 is Non-ablien group.
 Q8={< i, j>: i4
=j4
=1;i2
=j2
=-1; j.i.j=-i} (i.e The Group is generated by two
elements)
 Z(Q8)={1 , -1}
 Is not simple group.
Case-1

Subgroups:
8-Order Groups
Q8
H1=<i>={1 ,-1 ,i, -i} H2={1 ,-1 , j , -j}=<j> H3={1 ,-1, k ,-k}=<k>
H4={1 , -1}
H5={1}
 Every Subgroup of Q8 is Cyclic
subgroup.
 H , H ,H are isomorphic to (Z4 ,+4)
 Total numbers of subgroups are 6.
 Every subgroup of Q8 is normal
subgroup.
 Total normal subgroups of Q8 are 6

8-Order Groups
 D4={e ,a ,a2
, a3
, b , ba , ba2
,ba3
} is a group w.r.to multiplication.
 Properties:
 D4={<a,b> ; a4
=b2
=e , ab=ba3
}
 D4 is Non- abelian group.
 Z(D4)={e ,a2
}
 Is not Simple group
Caylay diagram
Case-2

D4
H1={e ,a ,a2
,a3
} H2={e, b,ba2
, a2
} H3={e , ba ,ba3
, a2
}
H4={e , a2
} H5={e ,ba} H6={e , ba2
} H7={e , b} H8={e ,ba3
}
H9={e}
 There are 4-Order subgroups are 3 and 2-Order sub groups are 5 of D4.
 Total Subgroups are 10.
 Every proper subgroup of D4 is cyclic expect H={e,b ,ba2
,a2
} ,
 H={e ,ba,ba3
,a2
} are abelian.
8-Order Groups
Subgroups:

Normal subgroups:
D4
H={e ,a ,a2
,a3
} H={e, b,ba2
, a2
} H={e , ba ,ba3
, a2
}
H={e , a2
}
H={e}
 4 proper subgroups are normal subgroups of D4.
 Total Normal subgroups are 6
8-Order Groups

 Z2XZ2XZ2 is an abelian Group of order 8 w.r.to +.Defined as fallows
(a , b ,c)+(d,e,f)=(a+2d , b+2e ,c +2 f)
+ (0,0,0) (0,0,1) (0,1,1) (1,1,1) (1,0,1) (1,1,0) (1,0,0) (0,1,0)
(0,0,0) (0,0,0) (0,0,1) (0,1,1) (1,1,1) (1,0,1) (1,1,0) (1,0,0) (0,1,0)
(0,0,1) (0,0,1) (0,0,0) (0,1,0) (1,1,0) (1,0,0) (1,1,1) (1,0,1) (0,1,1)
(0,1,1) (0,1,1) (0,1,0) (0,0,0) (1,0,0) (1,1,0) (1,0,1) (1,1,1) (0,0,1)
(1,1,1) (1,1,1) (1,1,0) (1,0,0) (0,0,0) (0,1,0) (0,0,1) (0,1,1) (1,0,1)
(1,0,1) (1,0,1) (1,0,0) (1,0,0) (0,1,0) (0,0,0) (0,1,1) (0,0,1) (1,1,1)
(1,1,0) (1,1,0) (1,1,1) (1,0,1) (0,0,1) (0,1,1) (0,0,0) (0,1,0) (1,0,0)
(1,0,0) (1,0,0) (1,0,1) (1,1,1) (0,1,1) (0,0,1) (0,1,0) (0,0,0) (1,1,0)
(0,1,0) (0,1,0) (0,1,1) (0,0,1) (1,0,1) (1,1,1) (1,0,0) (1,1,0) (0,0,0)
8-Order Groups Case-3

Caylay diagram:  Properties:
 G=Z2XZ2XZ2 is abelian group.
 Is not simple group
 Each element order is 2 other than
identity.
 Subgroups of Z2XZ2XZ2:
 2-order sub groups
 H={(0,0,0),(1,0,0)}
 H={(0,0,0) ,(0,1 ,1)}
 H={(0,0,0), (0,1,0)}
 H={(0,0,0),(1,0,1)}
 H={(0,0,0),(1,1,1)}
 H={(0,0,0),(1,1,0)}
 H={(0,0,0),(0,0,1)}
8-Order Groups

 4-order subgroups:
 W={(0,0,0),(0,1,1),(1,1,0),(1,0,1)}
 W={(0,0,0),(0,1,1),(0,1,0),(0,0,1)}
 W={(0,0,0),(0,1,1),(1,1,1),(1,0,0)}
 W={(0,0,0),(0,1,0),(1,1,0),(1,0,0)}
 W={(0,0,0),(0,0,1),(1,1,0),(1,1,1)}
 W={(0,0,0),(0,0,1),(1,1,0),(1,1,1)}
 W={(0,0,0),(0,0,1),(1,0,1),(1,0,0)}
 Every two order subgroup of Z2XZ2XZ2 is isomorphic to (Z2,+2 ,X2)
 Every 4-order subgroup of Z2XZ2XZ2 is isomorphic to Z2XZ2.
 Normal subgroups:
 Every sub group of Z2XZ2XZ2 is normal sub group
8-Order Groups

 Z4XZ2 is an abelian group of order 8 w.r.to + ,defined as
(a,b)+(c,d)=(a+2c ,b+2d) for all (a,b),(c,d) Zϵ 4XZ2
+ (0,0) (0,1) (1,0) (1,1) (2,0) (2,1) (3,0) (3,1)
(0,0) (0,0) (0,1) (1,0) (1,1) (2,0) (2,1) (3,0) (3,1)
(0,1) (0,1) (0,0) (1,1) (1,0) (2,1) (2,0) (3,1) (3,0)
(1,0) (1,0) (1,1) (2,0) (2,1) (3,0) (3,1) (0,0) (0,1)
(1,1) (1,1) (1,0) (2,1) (2,0) (3,1) (3,0) (0,1) (0,0)
(2,0) (2,0) (2,1) (3,0) (3,1) (0,0) (0,1) (1,0) (1,1)
(2,1) (2,1) (2,0) (3,1) (3,0) (0,1) (0,0) (1,1) (1,0)
(3,0) (3,0) (3,1) (0,0) (0,1) (1,0) (1,1) (2,0) (2,1)
(3,1) (3,1) (3,0) (0,1) (0,0) (1,1) (1,0) (2,1) (2,0)

 Properties:
 Z4XZ2 is an abelian ( Non-cyclic)
group of order 8.
 Is not simple group.
Caylay diagram :
Order of elements:
|(0,0)|=1
|(0,1)|=l.c.m[|0|,|1|]=2
|(1,0)|=l.c.m[|1|,|0|]=4
|(1,1)|=l.c.m[|1|,|1|]=4
|(2,0)|=l.c.m[|2|,|0|]=2
|(2,1)|=l.c.m[|2|,|1|]=2
|(3,0)|=l.c.m[|3|,|0|]=4
|(3,1)|=l.c.m[|3|,|1|]=4
8-Order Groups

Subgroups:
Z4XZ2
H1={(0,0),(1,0),(3,0),(2,0)} H2={(0,0),(1,1),(3,1),(2,0)} H3={(0,0),(2,0),(2,1),(0,1)}
H4={(0,0),(1,0)} H5={(0,0) ,(2,0)} H6={(0,0),(2,1)}
H7={(0,0)}
 Total number of sub groups is 8.
 Every 2-order subgroup is isomorphic (Z2,+2)
 Every 4-order subgroup is isomorphic to (Z4,+4) or Z2XZ2.
 Every subgroup of Z4XZ2 is normal
8-Order Groups

 Z8={0,1,2,3,4,5,6,7} is a cyclic group w.r.to +8 of order 8.
+ 0 1 2 3 4 5 6 7
0 0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7 0
2 2 3 4 5 6 7 0 1
3 4 5 6 7 0 1 2 3
4 5 6 7 0 1 2 3 4
5 6 7 0 1 2 3 4 5
6 7 0 1 2 3 4 5 6
7 7 0 1 2 3 4 5 6
Caylay diagram:
 Properties:
 (Z8 ,+8) is a cyclic group.
 Every 8-order cyclic group is isomorphic to Z8.

Z8
H={0,2,4,6}
H={0,4}
{0}
 If d|8 then there exist a unique subgroup of order d
 Number of total subgroups are 4.
 For each subgroup is cyclic.
Subgroups:
8-Order Groups

Chapter-2
HOMOMORPHISM OF GROUPS
Def:
let ‘G’ , ‘K’ are two Groups w.r.to O1 , O2 respectively. let ‘f’ be function
from G to K is said to be homomorphism.
If f (aO1b)=f(a)O2f(b) for all a,b Gϵ
a b
aO1b
f(a) f(b)
f(aO2b)
G K

Properties of Homomorphism
Let f:G---> K is homomorphism.
 f(e)=e|
where e , e|
are identity elements in G, k respectively.
 f(an
)=[f(a)]n
for all n Zϵ
 |f(a)| divides |a|, if |a|<∞
 Ker f={x G/ f(x)=eϵ |
} is called kernel of f, which is Normal sub
group of G.
 f(G)={f(x)/x G} is called Range of f, which is sub Group is of K.ϵ
G K
e e|
Ker f Range f

Properties of Homomorphism
 If |ker f|=1 f is one-to-one
 If |ker f|=2  f is Two –to-one mapping
 If |ker f|=3  f is Three –to- one mapping
 Fundamental Theorem of Homomorphism
 if H <G then f(H)={f(h) ; h H} is sub group of K.ϵ
 if H is normal sub group of G then f(H) is normal sub group of f(G).
 If N is normal sub group of K, then f-1
(N)={g G; f(n) N} is normalϵ ϵ
sub group of G.
 If f is bijective then G and K are isomorphic.

HOMOMORPHISAMS OF FINITE CYCLIC
GROUP TO FINITE CYCLIC GROUPS
Example: How many homomorphism’s from Z4 to Z6?
Sol: First we have to find normal sub group of Z4.
We know that every sub group of Z4 is Normal.
Let “f” is homomorphism from Z4 to Z6 then
|ker f|=1,2 (or) 4.
Let |ker f|=4
=> f is trivial homomorphism.
Let |ker f|=1
=> from (11) property 4|6
Is NOT true.
Let |ker f|=2
=> from (11) property 2|6.
. ‘. There is a possible a homomorphism form Z4 to Z6 which
is 2-to-1 mapping.

0
1
2
3
0
1
2
3
4
5
.’. The number of homomorphism is 2.

METHOD:
Let G , K are finite cyclic Groups of orders m , n respectively .

Example:
find all homomorphism from Z12 to Z14
Sol: The common divisors of 12 & 14 are 1, 2.
1-order elements are only one i.e 0.
Number Of 2-order elements in Z14 is one, that is 7.
.‘. Homomorphism’s from Z12 to Z14 are f(x)=0 and f(x)=7.x.
Total number of homeomorphisms is only two.

One-to-one homomorphism
Def: let H,K are two groups, f:HK is homomorphism, is said to be
one-to-one homo if f is one-to-one
Properties:
 f:HK is one-to-one homomorphism iff Ker f={e}
 H,K are finite groups, f:HK homo defined as f(x)=a.x for all
x H, where a K, if o(a)=o(H) then f is one-to-one.ϵ ϵ
One-to-one homomorphism from Cyclic group to cycly group:
 Let H,K are two finite order cyclic groups of orders m,n
respectively ,if m|n then there exist Ф(m) one-to-one
homomorphism from H to K.
 H,K are finite cyclic groups f:HK homo defined as f(x)=a.x for
all x H, where a K, if o(a)=o(H) then f is one-to-one.ϵ ϵ

Example
One-to-one homo from Z6 to Z12.
We have g.c.d(6,12)=6
That implies there exist 6 homo from Z6 to Z12.
And aslo 6|12,this implies there exist one-to-one homo from Z6 to Z12.
First we have to find all homo from Z6Z12.
The common divisors are of 6,12 are 1,2,3,6.
1-order elements in Z12 are only one, i.e 0
2-order elements in Z12 are only one ,i.e 6
3-order elements in Z12 are only two, they are 4,8
.’. Homomorphism are from Z6 Z12 are
f(x)=0x
f(x)=6x
f(x)=4x
f(x)=8x
f(x)=2x
f(x)=10x
we have the order of elements 2,10 in Z12 is 6.

On-to homomorphism
Def: let H,K are two groups, f:HK is homomorphism, is said to be
on-to homo if f is on-to
Properties:
f:HK is on-to homomorphism iff f(H)=K
On-to homomorphism from Cyclic group to cycly group
 Let H,K are two finite order cyclic groups of orders m,n
respectively ,if n|m then there exist Ф(n) on-to homomorphism
from H to K.
 H,K are finite cyclic groups f:HK homo defined as f(x)=a.x for
all x H, where a K, if o(a)=o(K) then f is on-to.ϵ ϵ

Example:
On-to homo from Z12 to Z6.
We have g.c.d(12,6)=6
That implies there exist 6 homo from Z12 to Z6.
And aslo 6|12, this implies there exist on-to homo from Z12 to Z6.
First we have to find all homo from Z12Z6.
The common divisors are of 6,12 are 1,2,3,6.
1-order elements in Z6 are only one, i.e 0
2-order elements in Z6 are only one ,i.e 3
.’. Homomorphism are from Z12 Z6 are
f(x)=0x
f(x)=3x
f(x)=2x
f(x)=4x
f(x)=1x
f(x)=5x
we have the order of elements 1,5 in Z6 is 6.

References:
Topic in algebra- I.N. Herstein
Joseph Gallian- Contemporary Abstract Algebra 2009
Abstract Algebra-JOHN BACHAY
P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul-Basic Abstract
Algebra-Cambridge University Press (1994)
The Theory of Groups - H. Bechtell
schuam- group theory
Group Explorer
GAP-joseph Gallaian

Classification of Groups and Homomorphism -By-Rajesh Bandari Yadav

Classification of Groups and Homomorphism -By-Rajesh Bandari Yadav

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