Linked Questions

Let $G$ be a finite group of order $n$ and $p$ be the minimal prime number dividing $n$. Assume that $H \subset G$ is a subgroup of index $p$. Prove that $H$ is normal: $H \trianglelefteq G$ To do ...

Assume that $G$ is finite with $p$ the smallest prime dividing its order. Suppose $H < G$ with $[G:H]=p$. Prove that $H \lhd G$. I've seen this question a few times on here but all the proofs I ...

Let $G$ be a finite group, $H \leq G$ such that $[G : H]$ is the least prime which divides $|G|$. Show that $H$ is normal in $G$. Can someone explain what information we get from knowing that $[G : H]...

Show that if a group $G$ has odd order, any subgroup $H$ of index 3 in $G$ is normal in $G$. I think this is equivalent to the following: Let $H$ and $K$ be subgroups of a group $G$, with $K \leq H$...

Basically what the title says. If $|G|=27$ for example, how would one prove that $N$ is normal in $G$ if $|N|=9$

If $|G|=p^{n}$ Then Why is it that every subgroup of order $p^{n-1}$ is normal?

This is a problem from Herstein that I have been stuck upon for ages. I am becoming increasingly disappointed and disillusioned about my abilities due to this problem. Let G be finite and let $p$ ...

It is known that any subgroup of a group having index 2 is normal. Note that here 2 is a prime. Is such a result true for some other odd prime ?. That is, "does there exist an odd prime $p$ such ...

Let $G$ be a group and $H$ a non-normal subgroup of $G$. Please prove that the order of $G$ is divided by a prime number which is less than $[G:H]$.

Let $G$ be a finite group, and let $H$ be a subgroup of $G$ such that the index $(G\colon H)$ of $H$ in $G$ is the smallest prime that divides the order of $G$. Can we say anything about whether or ...

Since $p$ is the smallest prime dividing $|G|$, and $|G:H|=p$, we know that $H$ is the largest proper subgroup of $G$. So the normalizer $N_G(H)$ must be equal to either $G$ or $H$. If $N_G(H)=G$, ...

Given that $H\leq G$ of index $p$ in $G$, where $p$ is the smallest prime integer such that $p \mid |G|$, then $H \trianglelefteq G$. I would appreciate some hints, as I don't even know where to begin....

If $G$ is a finite group of order $n$ and $p$ is the smallest prime dividing $|G|$, then prove that any subgroup of index $p$ is normal. This falls under the category of Cayley's Theorem, but i have ...

I am struggling to understand this proof. Suppose that $p$ is the smallest prime that divides $|G|$ show that any subgroup of index $p$ is normal in $G$. Proof Let $\phi \rightarrow S_p$ be the ...

Let $H\le G$ and suppose that $[G:H]$ is the least positive prime factor of $|G|$. How to show that $H\triangleleft G$? I am confused in how to establish relation between the number of cosets of $H$ ...