Determine order of $\langle(12345),(2354)\rangle$.

Question

I'm asked to prove that the order of $H:=\langle(12345),(2354)\rangle$ is $20$, in an Algebra exercise 2nd year maths.

What I have achieved is that $20\mid |H|$ and $|H|=20,120$ since $H$ can't be the alternating group and there are no subgroups of order 40 in $S_5$. But I don't know how to prove $H\neq S_5$.

Thanks!

Answer 1

Consider the integers modulo $5$. Let $\alpha$ be the operation $\alpha\colon x\mapsto x+1$. Let $\beta$ be the operation $\beta\colon x\mapsto 2x$.

Then $\alpha,\beta$ generate all affine automorphisms of $\mathbb{Z}/5\mathbb{Z}$, which has size $20$: $$x\mapsto ax+b,$$ with $a\in \{1,2,3,4\}$ and $b\in \{0,1,2,3,4,5\}$.

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As pointed out by @JeanMarie, we have shifted the indices by $-1$ here, so $\alpha$ is the permutation $(01234)$ and $\beta$ is the permutation $(1243)$.