# Questions tagged [p-groups]

Use with the (group-theory) tag. Refers to questions concerning finite groups of prime power order or infinite p-groups such as Prüfer groups, pro-p-groups, and Tarski monsters. This tag is not for p-adic number systems.

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### Let $G$ be finite $p$-group whose set of irreducible character degrees is $\{1,p^a\}$ and $H\unlhd G$ s.t. $G/H$ is nonabelian. Is $H\le Z(G)$? [closed]

Let $G$ be finite $p$-group whose set of irreducible character degrees is $\{1,p^a\}$ and $H$ is a normal subgroup of $G$ such that $G/H$ is a nonabelian group. May I assume that $H\le Z(G)$?
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### $p$-groups containing normal subgroup of orders of powers of $p$. [closed]

Let $G$ be a $p$-group such that $|G| = p^e$ where $p$ is prime and $e \geq 1$. Prove by induction that $G$ contains a normal subgroup of every possible order $p^i$ for $i = 1,\ldots, e-1$. I am ...
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### Is every direct product of (finite) cyclic groups abelian?

Long story short I'm halfway through a proof and have hit a step where I show that a group order $p^2$ is abelian. I did this by splitting into the $C_{p^2}$ case and the $C_p\times C_p$ case, the ...
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### Let $p$ be a prime number and let $G$ be a $p$-group. Then $G$ contains a normal subgroup of order $p^k$ for every nonnegative $k\le r$

Let $p$ be a prime number, and let $G$ be a $p$-group: $|G|=p^r$ . Then $G$ contains a normal subgroup of order $p^k$ for every nonnegative $k\le r$ But are there any normal subgroup of order $p^n$ ...
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I'm trying to solve this group theory problem from my abstract algebra course. It goes like this: Prove that any group of order 945 has at least one subgroup of order 9. First, I noticed that $945=3^... 2answers 84 views ### Prove that any group of order$27$is not simple I'm stuck on this problem from my abstract algebra course: Prove that if$G$is a group with$|G|=27$, then$G$is not simple. First I noticed$|G|=27=3^3$. I thought I can use a statement I saw on ... 0answers 46 views ### Construction of an automorphism of elementary abelian p-group of order$p^{2}$Let$G$be an elementary abelian p-group of order$p^{2}$. Let$g$,$h$be two non-trivial elements of$G$. I want to construct an automorphism$\phi\in Aut(G)$such that$\phi(h)=g$, which is ... 1answer 95 views ### Prove that any group$G$with$|G|=588$is solvable I'm stuck trying to solve this problem from my abstract algebra course: Prove that every group of order$588$is solvable (If you assume that all groups of certain order are solvable, you must prove ... 1answer 45 views ### All 2-groups have cyclic commutator subgroups? I have seen that groups of order 8 and order 16 have commutator subgroup cyclic. I want to know if that can be extended to all 2-groups or if there is a reason why this happens. 1answer 57 views ### How to prove the existence of non-abelian group of order$125$The following question was part of my algebra assignment. Let$G$be a non abelian group. Can its order be$125 ?p$groups have non-trivial center. So, if$|Z(G)| =125$and the group is abelian, ... 1answer 182 views ### Conjugacy classes of a nonabelian group of order$p^4$Apologies in advance for the long text! I will explain what I have done so far, and then I will present my question. Consider a nonabelian group of order$p^4$, where$p$is a prime. Then, the center$...
Let $G$ be a finite non-abelian group, and lets randomly choose two elements of $G.$ It seems quit well known that the probability that they commute is at most $\text{Pr}(G)\le\dfrac{5}{8}.$ Here is a ...