# 2-Sylow subgroup of symmetric group

## Definition

### As a Sylow subgroup

The term 2-Sylow subgroup of symmetric group refers to a group that occurs as the 2-Sylow subgroup of a symmetric group on finite set, i.e., a symmetric group on a set of finite size.

For every natural number $n$, there is a corresponding 2-Sylow subgroup of the symmetric group $S_n$. For an even number and the odd number bigger than it by 1, the 2-Sylow subgroups of the corresponding symmetric groups are isomorphic.

## Particular cases $n$ symmetric group $S_n$ order of $S_n$ (= $n!$) 2-Sylow subgroup order of 2-Sylow subgroup (= largest power of 2 dividing $n!$) log to the base 2 of order of 2-Sylow subgroup (= sum of the integer parts of $n/2^k$, $k$ varying over positive integers) Information on Sylow subgroup inside symmetric group
0 trivial group 1 trivial group 1 0 --
1 trivial group 1 trivial group 1 0 --
2 cyclic group:Z2 2 cyclic group:Z2 2 1 --
3 symmetric group:S3 6 cyclic group:Z2 2 1 S2 in S3
4 symmetric group:S4 24 dihedral group:D8 8 3 D8 in S4
5 symmetric group:S5 120 dihedral group:D8 8 3 D8 in S5
6 symmetric group:S6 720 direct product of D8 and Z2 16 4
7 symmetric group:S7 5040 direct product of D8 and Z2 16 4
8 symmetric group:S8 40320 wreath product of D8 and Z2 128 7
9 symmetric group:S9 362880 wreath product of D8 and Z2 128 7