# Does #S_5# have a subgroup of order #40#?

##### 1 Answer

I don't have a complete answer for you, but here are a few thoughts...

#### Explanation:

By Lagrange's Theorem, any subgroup of

The converse is not true. That is, if a group has order a factor of

It may help to look at what the possible generators of a subgroup of order

(a)

#" "(1, 2)" "# i.e. one adjacent transposition(b)

#" "(1, 3)" "# i.e. one non-adjacent transposition(c)

#" "(1, 2)(3, 4)" "# i.e. two adjacent transpositions(d)

#" "(1, 3)(2,4)" "# i.e. two non-adjacent transpositions(e)

#" "(1, 2)(3, 5)" "# i.e. one adjacent, one non-adjacent transpositions

Combined with

(a)

#" "S_5" "# order#120# (b)

#" "S_5" "# order#120# (c)

#" "A_5" "# order#60# (d)

#" "D_5" "# order#10# (e) ?

Anyway, we can look through possible generators and equivalences, hence enumerating and excluding possibilities.