49 questions linked to/from Normal subgroup of prime index
3k views

### $G$ is a finite group such that $|G| = n$ and $p$ is minimal prime dividing $n$. $H \subset G, [G:H] = p$. Prove $H$ is normal in $G$ [duplicate]

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 ...
2k views

### If a subgroup has smallest prime index, then it is normal [duplicate]

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 ...
678 views

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 $[G : H]... 1answer 954 views ### Show that if a group$G$has odd order, any subgroup$H$of index 3 in$G$is normal in$G$? [duplicate] Show that if a group$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$... 1answer 632 views ### Proving that for$|G|= p^n$, that some subgroup$N$is normal in G, where$|N|=p^{n-1}$[duplicate] Basically what the title says. If$|G|=27$for example, how would one prove that$N$is normal in$G$if$|N|=9$1answer 794 views ### Why is every subgroup of order$p^{n-1}$normal? [duplicate] If$|G|=p^{n}$Then Why is it that every subgroup of order$p^{n-1}$is normal? 0answers 458 views ### Let G be finite and let$p$be the smallest prime dividing$|G|$. Let$H \le G$be of index$p$. Prove that$H$is a normal subgroup of$G$. [duplicate] This is a problem from Herstein that I have been stuck upon for ages. I am becoming increasingly disappointed and disillusioned about my abilities due to this problem. Let G be finite and let$p$... 2answers 280 views ### normality of subgroup of odd prime index [duplicate] It is known that any subgroup of a group having index 2 is normal. Note that here 2 is a prime. Is such a result true for some other odd prime ?. That is, "does there exist an odd prime$p$such ... 1answer 206 views ### Non-normal subgroup [duplicate] Let$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]$. 1answer 121 views ### Is this type of a subgroup always normal? [duplicate] Let$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$. Can we say anything about whether or ... 1answer 127 views ###$|G|<\infty$.$p$the smallest prime dividing$|G|$.$|G:H|=p$. then H is normal. [duplicate] Since$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$, ... 0answers 96 views ### If$H\leq G$of index$p$in$G$and$p\mid |G|$then$H\trianglelefteq G$[duplicate] Given that$H\leq G$of index$p$in$G$, where$p$is the smallest prime integer such that$p \mid |G|$, then$H \trianglelefteq G$. I would appreciate some hints, as I don't even know where to begin.... 2answers 62 views ### How do i show any subgroup of index$p$is normal? [duplicate] If$G$is a finite group of order$n$and$p$is the smallest prime dividing$|G|$, then prove that any subgroup of index$p$is normal. This falls under the category of Cayley's Theorem, but i have ... 1answer 27 views ###$p$smallest prime dividing$G$subgroup with index$p$normal in$G$[duplicate] I am struggling to understand this proof. Suppose that$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$be the ... 0answers 22 views ### About coset and normal subgroup [duplicate] Let$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\$ ...

15 30 50 per page