International Journal of Mathematics Trends and Technology (IJMTT) – Volume 37 Issue 2- September 2016
On Embedding of Every Finite Group into a
Group of Automorphism
#2
Kuldeep Singh#1, Ankur Bala#2
Department of Mathematics, Government PG College, Hisar (Haryana)
Department of Applied Science, CDL Government Polytechnic College (Nathusari Chopta)
India
#1
Abstract— In this Article, We have proved that every
group of finite order can be embedded into a group of
automorphism. We have used the famous classic result
of cayley, which states that every group can be
embedded into a group of permutations.
Aut      
 n copies 
Keywords— Embedding, Finite Group, Group of
Automorphism,
 n copies  is
Aut ( n ) = Aut      
Notation:
group of automorphisms with respect to composition
of function.
GL( n, ) 

[aij ]nn : aij  
&
det( [aij ]nn ) = 1

Theorem 1: (Cayley’s ) Every finite group can be
embedded in S n for some n  N .
to Aut
GL( n, ) is isomorphic
 n copies  .
     
Theorem 2:
Theorem 3:
Sn can be embedded in GL( n, ) for
all n  N .
Main Theorem: Using theorem 1, 2 &3 we can say
that every finite group can be embedded into group
Aut      
  n copies  for
some n  N .
Proof of Theorem 2:
Let f , g  Aut (
Then
n
)
f (0,1, 0,0....)  ( a21 , a21 ,.....a2 n )
f (0, 0, 0,....,1)  (an1 , an 2 ,....., ann )
Now define
 : Aut ( n ) 
 GL( n,  )
ISSN: 2231-5373
automorphism which is multiplication by
 b11  b1n 


N     
b  b 
nn 
 n1
Such that MN  I .
The
determinants
are
integers
satisfying
det( M ).det( M 1 )  1
 det( M )  1
 There is a bijection between GL( n, )
and Aut ( ) .
Composition of automorphisms corresponds to
multiplication of matrices.
So, it is an isomorphism.
n
 Aut ( n ) is isomorphic to GL( n, ) .
Proof of Theorem 3:
Let Sn be the permutation group on n symbols.
 : S n 
 GL  n,  
 ( )   nn   S n
.
.
 a11  a1n 
 ( f )      
a  a 
nn 
 n1
 f is associated with a n  n matrix with
determinant 1 .
If f is an automorphism then f has an inverse
Define
f (1, 0, 0,0....)  ( a11 , a12 ,.....a1n )
.
Such that
Where
.
i.e. if
such that:
 nn is permutation matrix obtained by
1
2
n
 

 1  2   n 

then
 R1 
 
R
 nn   2 
 
 Rn 
th
Where Ri is R row of identity matrix.
i
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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 37 Issue 2- September 2016
Clearly  is a homomorphism.
Now consider the kernel of this homomorphism.
ker    :  ( )  I nn 
 i  i i
 ker  is trivial.
Hence the homomorphism is injective.
 : Sn can be embedded in GL( n, ) for
all n  N .
Proof of Main Theorem:
Since every finite group can be embedded in Sn as
stated in theorem 1 & by theorem 3 we can say that
Sn can be embedded in GL( n, ) .
So, every finite group can be embedded in
GL( n, ) .
And in theorem 2, we have proved that
is isomorphic to
Aut ( ) .
ISSN: 2231-5373
n
GL( n, )
Now, combining all these statements one can easily
conclude that every finite group is isomorphic to a
subgroup of a group of automorphisms.
CONCLUSION
Every finite group is isomorphic to a subgroup of
group of permutations by the famous cayley’s
theorem.
But we have proved that every finite group is
isomorphic to the subgroup of a group of
automorphisms i.e.
[1]
[2]
Aut ( n ) .
REFERENCES
Contemporary Abstract Algebra, J.A. Gallian, Cengage
Learning, 2016 .ISBN: 1305887859, 9781305887855
Abstract Algebra (3rd Edition), David S. Dummit , Richard
M. Foote, ISBN: 978-0-471-43334-7
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