# In S4, what is the subgroup generated by the cycle (123)?

Let $$S_4$$ be the symmetric group of degree $$4$$ and $$H$$ the subgroup of $$S_4$$ generated by $$(1\ 2\ 3)$$. I want to list out the members of $$H$$.

I know they are the powers of (123), but I get (132) when I raised (123) to power of 2 which seems to be the same thing as (123). Does it mean the subgroup has only one member (123)?

Also what is the quotient group $$S_4/H$$?

• $H$ is not a normal subgroup of $S_4$, so there is no quotient group $S_4/H$. – Derek Holt Jun 2 at 17:50

No: $$(1\ \ 2\ \ 3)\neq(1\ \ 3\ \ 2)$$, since $$(1\ \ 2\ \ 3)$$ maps $$1$$ into $$2$$, whereas $$(1\ \ 3\ \ 2)$$ maps $$1$$ into $$3$$.
You can also check that $$(1\ \ 2\ \ 3)^3=e$$. Therefore,$$H\left(=\bigl\langle(1\ \ 2\ \ 3)\bigr\rangle\right)=\bigl\{(1\ \ 2\ \ 3),(1\ \ 3\ \ 2),e\bigr\}.$$