- #1

- 147

- 0

## Main Question or Discussion Point

Is it true that if a finite group G contains a subgroup of index 2, then there is an element of G with order 2?

- Thread starter ForMyThunder
- Start date

- #1

- 147

- 0

Is it true that if a finite group G contains a subgroup of index 2, then there is an element of G with order 2?

- #2

- 147

- 0

- #3

- 348

- 0

Regarding your proof: Is multiplication of a left coset by a right coset well defined? And why do you end up with H on the right side? Doing this assumes xx=e.

- #4

- 147

- 0

But you have to have x^2 in H for all x not in H. (xH)(xH)=H because otherwise, it would be an identity: (xH)(xH)=(xH) and cancellation gives that x is in H.

I don't see any way to prove this.

- Last Post

- Replies
- 2

- Views
- 2K

- Last Post

- Replies
- 3

- Views
- 3K

- Last Post

- Replies
- 1

- Views
- 1K

- Last Post

- Replies
- 1

- Views
- 1K

- Last Post

- Replies
- 4

- Views
- 2K

- Last Post

- Replies
- 1

- Views
- 2K

- Last Post

- Replies
- 6

- Views
- 3K

- Replies
- 2

- Views
- 2K

- Replies
- 8

- Views
- 4K

- Last Post

- Replies
- 3

- Views
- 4K