Sylow Prime Group

Sylow Prime Group

T. Srinivasarao Asst. Professor Dept. of Math

Mr. K. Revathi Asst. Professor Dept. of Math

University College of Science & Technology Adikavi Nannaya University

University College of Science & Technology Adikavi Nannaya University

Abstract: Sylow p subgroup is a vital part of the discussion in any algebraic activity dealing with group theory. So, it is natural that, is every group can have Sylow p subgroups or any specific distinction can be put forward that confirms a particular group is either a group having Sylow p subgroup or there is no single Sylow p subgroup. With a view to characterize the groups whether possessing a Sylow p subgroup or not here is an activity that progresses the discussion by one step ahead. This discussion leads to groups of order p and order pq that has applications in Galvan theory.

INTRODUCTION:

Suppose n is a positive integer. By the fundamental theorem of arithmetic, either n is a group of prime order or it is a product of

primes expressible in a unique manner as n p 1 p 2 … p k

where

p ' s are prime numbers with the respective multiplicities

i ,1 i k .

1 2 k i

Since each p is a prime number, 1 i k , there exists a cyclic group G ,1 i k such that

G p i ,1 i k

i

Now, G G1 G2 … Gk

i i i

1. Sylow Prime Group:

   Definition : if G is a finite group, p is a prime number such that called a Sylow p subgroup of G. (1.1)

   pn | G and pn1 | G , then any subgroup of G of order

   pn is

   Definition 2: a group (G, *) is said to be a Sylow prime group if every non trivial subgroup of G is a Sylow

   pi – subgroup for

   some prime factor (1.2)

   pi of the order of G.

   Since G n p 1 p 2 …p k , and in view of Lagranges theorem of finite groups, and properties of divisibility, it follows that

   1 2 k

   p i | G

   for each 1 i k and p

   i1 | G

   by the unique representation of the integer n.

   i i1

   pi i

   pi i

   It is not necessary that there is a subgroup of G of order for every 1 i k . This confirms that every group of finite order is not a Sylow prime group.

   To verify these observations, the following instances will show a finite group that admits the definition of Sylow prime group and another instance for not.

2. Working on Sylow Prime Groups:

Consider the symmetric group of order 6 or the symmetric group on 3 symbols.

S3 f1, f2 , f3 , f4 , f5 , f6 where

f1 a, a,b,b,c,c f2 a,b,b, a,c,c f3 a, a,b,c,c,b f4 a,c,b,b,c, a f5 a,b,b,c,c, a f6 a,c,b, a,c,b

fi : A A is a bijection for each 1 i 6 and

A a,b,c

The composition of mappings is the operation that makes S3 a group such that S 6 2 3 , the unique representation

1 1

3

3

3

3

by fundamental theorem of arithmetic

It can be easily seen that H f , f , H f , f , f are the only non trivial subgroups such that H 21,22 | S and so,

1 1 4 2 1 5 6 1 3

H1 is a Sylow 2- subgroup of S3

Similarly, H 31,32 | S and so, H is a Sylow 3 subgroup of S (2.1)

2 3 2 3

Take another instance.

12 1,5,7,11is a group under multiplication modulo 12 denoted by 12 .

12 4 22

H1 1,11 is a non trivial subgroup of order 21 .

1

1

Also, 22 | 12 which shows H is not a Sylow 2 subgroup of 12

So, this is an example of a group that is not a Sylow prime group. (2.2) The working (2.1) and (2.2) will confirm that all finite groups are not Sylow Prime Groups.

REFERENCES:

1. R.Gow, Sylows proof of Sylows theorem, Irish Math.Sco.Bull. (1994), 55 63

2. L.Sylow, Theorems sur les groups de substitutions, Mathematische Annalen 5(1872), 584 594

3. W. C. Waterhouse, The early proofs of Sylows theorems, Arch.Hist.Exact Sci, 21(1979/80), 279-290

4. William Fulton and Joe Harris. Representation Theory: A First Course. GTM 129, Springer, 1991