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I have an exam type question on subgroups/index, but I am completely lost on what to do as I don't really understand the whole idea of cosets:

If G is a finite group with subgroups H and Z, show that:

$[G:H\cap Z]=[G:H]\cdot[H:H\cap Z]\leq [G:H]\cdot[G:Z]$ $\\$

Any help would be greatly appreciated. Thanks

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  • $\begingroup$ I am sure this was asked here so just search Jessica. (-: $\endgroup$ – mrs Mar 3 '14 at 13:41
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The first equality is evident. Hence if you can show that $[H:H \cap Z] \leq [G:Z]$, then you are done. Define a map $\phi: \{H$-cosets of $H \cap Z\} \rightarrow \{G$-cosets of $Z\}$ by $\phi(x(H \cap Z))= xZ$, for an $x \in H$. Then $\phi$ is well-defined: if $x(H \cap Z)=y(H \cap Z)$, then $y^{-1}x \in H \cap Z$, hence $y^{-1}x \in Z$, so $\phi(x)=\phi(y)$, that is, $\phi$ does not depend on the particular representative of the coset. Next, you want to show that $\phi$ is an injective map. I leave that to you, can you take it from here?

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