# 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.

354 questions
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### Number of conjugacy classes of a finite p-group

Let $p$ be a prime number and $G$ be a non-Abelian finite $p$-group. Is it true that the number of conjugacy classes of $G$ is not a power of $p$?
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### Index of subgroups in infinite p-group

If $G$ is a finite $p$-group, it is trivial that every subgroup has index $p^r$ for some integer $r$. If $G$ is infinite, this is not true as the index can be infinite. If $G$ is an infinite $p$-...
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### Bounding the exponent in unit groups of modular group algebras

Let $p$ be a prime number, $G=:G_0$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_1:=1+\operatorname{rad}(KG)$ is a p-group ...
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### Isomorphism problem for the center of modular group algebras

Let $p$ be a prime number, $G,H$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$. My ...
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### Defect of subnormality in modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$. $G$ is ...
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### Let $G$ be a group of order $p^n$ where $p$ is a prime and $n \in \mathbb{N}$. Prove that exist normal subgroups [duplicate]

Let $G$ be a group of order $p^n$ where $p$ is a prime and $n \in \mathbb{N}$. Prove that exist normal subgroups $N_{1},N_{2},N_{3},...,N_{n}$ with $|N_{i}|=p^i$ for all $i \in${$1,2,3,...,n$}.
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### Prove, that group of order $p^2$ is abelian.

I know there is a proof using these theorems: The center of a finite p−group is non-trivial For any group G , $G/Z(G)$ is cyclic iff $G$ is abelian, or in otherwords: the quotient $G/Z(G)$ can never ...
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### Does non-abelian $\Phi(G)$ imply, that $p^5 | |G|$ for some prime $p$?

Suppose $G$ is a finite group, such that $\Phi(G)$ is non-abelian. Does there always exist such prime $p$, that $p^5 | |G|$? Here $\Phi$ stands for Frattini subgroup. Using the same method, as the ...
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### Rank of a $p$-group

Here I've found two definitions of the rank of a $p$-group https://groupprops.subwiki.org/wiki/Rank_of_a_p-group. However, for the $2$-group $\mathbb{Z}/\mathbb{Z}_{4}$, the rank with the first ...
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### $p$-group problem

Let $A,B,C$ are three subgroups in a way that $1<A \triangleleft B \triangleleft C$. With $B/A$ and $C/B$ are $p$-groups. Then prove that $|C|$ is also a $p$-group. I have been trying to prove it ...
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### Cauchy's theorem proof clarification (group theory)

Cauchy's theorem says that if $G$ is a finite group with $p | |G|$ (when $p$ is prime), then $G$ contains an element of order $p$. When following the proof from wikipedia: https://en.wikipedia.org/...
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### Order of elements of the Prüfer groups $\mathbb{Z}(p^{\infty})$ [duplicate]

Let $\mathbb{Z}(p^{\infty})$ be defined by $\mathbb{Z}(p^{\infty}) = \{ \overline{a/b} \in \mathbb{Q}/ \mathbb{Z} / a,b \in \mathbb{Z}, b=p^i$ $with$ $i \in \mathbb{N} \}$, I wish show that any ...
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### In a group $G$ where $\exists!$ nontrivial, proper subgroup, show that $G$ is cyclic and $\lvert G\rvert=p^2$, for $p$ prime.

Suppose that $G$ is a group that has exactly one nontrivial proper subgroup. Then we have to show that $G$ is cyclic and order of $G$ is $p^2$ where $p$ is prime. I tried as, if $a$ and $b$ two ...
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### If $G$ is a $p$-group then $\Phi(G)=G'G^p$

Okay this problem is quite the confounding one for me. If $G$ is a $p$-group then it follows that $\Phi(G)=G'G^p$. Where: $\Phi(G)$- Frattini subgroup (which in this case is the intersection ...
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### If $\operatorname{class}(G) = 2$ and $\exp(G) = 4$ then $\exp(G') = 2$?

Let $G$ be a finite $p$-group. I'd like to prove (or disprove) that if the nilpotency class of $G$ equals two (i.e., $1 \neq G' \le Z$, where $Z$ is the center of $G$) and the exponent of $G$ equals ...
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### Let $G$ be a non-abelian finite $p$-group, do we have $Z(G) \leq \Phi(G)$ in general?

Let $G$ be a non-abelian finite $p$-group. I wonder if $Z(G) \leq \Phi(G)$, where $\Phi(G)$ is the Frattini subgroup of $G$? I'd known that it is true for non-abelian groups of order $p^3$ but I ...
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### Finding the number of subgroups of $\mathbb{Z}_{p^3} \oplus \mathbb{Z}_{p^2}$.

What is the number of subgroups of $\mathbb{Z}_{p^3} \oplus \mathbb{Z}_{p^2}$? I'm not really sure how to figure it out. I tried seeing subgroups of each $\Bbb Z_{p^n}$ but I'm not sure I'm going to ...
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### If $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$

In 'Finite Groups' by Gorenstein, it is stated that if $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$′. The proof is the ...
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### Relationship between Frattini subgroup and the first Agemo subgroup of a prime-power order group

Let $G$ be a $p$-group for an odd prime $p$. The Frattini subgorup $\Phi(G)$ is defined as the intersection of all maximal subgroups or, equivalently, as the subgroup of non-generating elements, i.e. ...
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### Bound on order of commutator subgroup of a $p$-group

I was reading an article where it is claimed that If $G$ is a finite $p$-group with $|G|=p^n$ and nilpotency class of $n-2$ where $n\ge 7$ then $p\le|Z(G)|\le p^2$ and $p^{n-3}\le |G'|\le p^{n-2}$....
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### Any $p$-order subgroup is normal in a $pk$ group

I'm looking to prove that if $G$ is a group of order $pk$ where $p$ is prime and $p>k$, that any subgroup $K\leq G$ of order $p$ is normal in G. Does anybody have any hints or tips for proving ...
### Number of normal subgroups of order $p^s$ of a $p$-group
Let $G$ be a $p$-group. Show that the number of normal subgroups of $G$ that have order $p^s$ is $1$ $mod(p)$. I think I have to use Sylow theorems and the fact that every subgroup of order $p^s$ is ...