Random Notes

[infimath-blog]

Applying Sylow theorem for finding Normal Subgroup

Sylow Theorem: Let |G|=p^{\alpha}m, for some prime p such that p^{\alpha +1} \nmid |G|. If n be the number of p- Sylow subgroup(subgroup of order p^{\alpha}), then it will satisfy the following condition

n \mid m, and

n = 1 ~(\!\!\!\mod p).

Any subgroup of unique order is normal.

Classification of group of order pq: If G is a group of order pq, for primes p<q then we can have following possibilities

If p \nmid (q-1), then only cyclic group of order pq is possible.

If p \mid (q-1), then we get one cyclic group of order pq and one non-abelian group of order pq

G is a finite group of order p^2q where in p and q are distinct primes such that q \nmid (p^2-1) and p \nmid (q-1). Then G is an abelian group.

[infimath-blog]

How to write class equation of Dihedral Group

Class equation of dihedral group, D_{2n}, is given by,

when n is odd, i.e., n =2m+1, 2 n=1 + \underbrace{2 +2+ \cdots +2}_{m \rm{\,times}} +n,

when n is even, n = 2m, 2n = 1+1+ \underbrace {2 +2 +\cdots +2 }_{m-1 ~\rm{ times}} +m +m.

[infimath-blog]

Permutation Group S3 and S4.

Elements of S_3

The symmetric group on three letters S_3 has 6 elements.

Cycle Structure Elements Even/Odd Identity (~) Even All two cycle. (12),(13),(23) Odd All three cycle (123),(132) Even

Note: The 3 order subgroup generated by (123) is a normal subgroup of S_3 and denoted by A_3.

Elements of S_4

The symmetric group on three letters S_3 has 6 elements.

Cycle Structure Elements Even/Odd Identity (~) Even All 2-cycles (12),(13),(14),(23),(24),(34) Odd Product of two 2-cycles (12)(34),(13)(24),(14)(23) Even All 3-cycle (123),(132),(124),(142),(134),(143),(234),(243) Even All 4-cycle (1234),(1432),(1324),(1423),(1243),(1342) Odd

Note 1: All even elements of S_4 form a normal subgroup of order 12 is known as A_4.

Note 2: Product of disjoint two 2-cycle with identity forms a normal subgroup of order 4 in S_4. It will be automatically normal in A_4.