Generalizing the case $p=2$ we would like to know if the statement below is true.
Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal.
Thanks.
Tell me more
×
Mathematics Stack Exchange is a question and answer site for
people studying math at any level and professionals in related fields. It's 100% free, no registration required.
|
|
|
This is a standard exercise, and the answer is that the statement is true, but the proof is rather different from the elementary way in which the $p=2$ case can be proven. Let $H$ be a subgroup of index $p$ where $p$ is the smallest index that divides $|G|$. Then $G$ acts on the set of left cosets of $H$, $\{gH\mid g\in G\}$ by left multiplication, $x\cdot(gH) = xgH$. This action induces a homomorphism $G\to S_p$, whose kernel is contained in $H$. Let $K$ be the kernel. Then $G/K$ is isomorphic to a subgroup of $S_p$, and so has order dividing $p!$. But it must also have order dividing $|G|$, and since $p$ is the smallest prime that divides $|G|$, it follows that $|G/K|=p$. Since $|G/K| = [G:K]=[G:H][H:K] = p[H:K]$, it follows that $[H:K]=1$, so $K=H$. Since $K$ is normal, $H$ was in fact normal. |
|||
|
Hint: Consider the set of cosets $G/H$ of which there are $p$. Then $G$ acts on these cosets by left multiplication so you have a homomorphism $\phi: G \rightarrow S_p$. If $p$ is the smallest prime dividing $|G|$ then what can you say about $|\mathrm{im} \phi|$ and what does this imply about $\ker \phi$? |
|||
|
|
Here is a slightly different way to prove the result: We will do it by induction on $|G|$. If $G$ has just one subgroup of index $p$ then clearly that subgroup is normal, so let $H_1$ and $H_2$ be distinct subgroups of index $p$. We then have that $|H_1H_2|$ is a multiple of $|H_1|$, but due to the choice of $p$ we must in fact have $H_1H_2 = G$ which means that if we let $K = H_1 \cap H_2$ then $K$ has index $p$ in $H_1$ and $H_2$ so by induction we know that $K$ is normal in $H_1$ and $H_2$ and thus normal in $G$. Now we know that $G/K$ has order $p^2$ so it is abelian. Now since both $H_1$ and $H_2$ contain $K$ they correspond to subgroups of $G/K$ and since this is abelian, they correspond to normal subgroups, which shows that they are normal in $G$ as desired. |
|||
|
|
|
Since $[G: H]=2$, the group $G$ has only two distinct left cosets and only two distinct right cosets. Now $H$ itself is a left as well as a right coset in $G$. Let $a \in G$. If $a \in H$, then $$aH=H=Ha$$. Suppose $a$ does not belong to $H$.Then $aH \neq H$. Hence $G=H \cup aH$ and $H \cap aH= \varnothing$. Then $aH=G-H$. Since $a$ does not belong to $H$ and $G$ has only two right cosets, we find that $G= H \cup Ha$ where $H \cap Ha= \varnothing $. Thus $Ha=G-H$. Hence $Ha=aH$.Thus we find that $aH=Ha$ for all $a \in G$ and so $H$ is a normal subgroup of $G$. |
|||||
|
|
Hint: Let $G$ act on $G/H$ by left multiplication. This gives you a homomorphism $G\to S_p$. Try to show that $H$ is the kernel of this map--note that if $q$ is a prime larger than $p$ then $q\nmid p!$. |
|||||||
|
|
Hint: I think you should try to work with what Alex suggested. It is usually referred to as the "Strong Cayley Theorem". |
||||
|
|
