- #1

DeldotB

- 117

- 7

## Homework Statement

Good day,

I need to show that [itex]S_n=\mathbb{Z}_2[/itex](semi direct product)[itex]Alt(n)[/itex]

Where [itex] S_n[/itex] is the symmetric group and [itex]Alt(n)[/itex] is the alternating group (group of even permutations) note: I do not know the latex code for semi direct product

## Homework Equations

none

## The Attempt at a Solution

(I think) It suffices to show that [itex]\mathbb{Z}_2\cap Alt(n)=0[/itex] (where 0 is the identity) and that

[itex]S_n= \mathbb{Z}_2 Alt(n)[/itex]. My question is, is [itex]S_n= \mathbb{Z}_2 Alt(n)[/itex] even valid notation? And how do I begin to do this? p.s I get this notation from my book which says:

To show a group G is a semi direct product, show [itex]G=NH[/itex] and [itex]N \cap H[/itex]= identity.

I should mention here that the alternating group is the normal subgroup (I think).

For the first part, since [itex] \mathbb{Z}_2[/itex]= {0,1}, its pretty clear that its intersection with Alt(n) is just the identity. I am having problems with the second part..