# 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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### Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization of the groups in which ...
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### More than 99% of groups of order less than 2000 are of order 1024?

In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024. Is this for real? How can one deduce this result? ...
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### A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$

This is problem 3 from Hungerford's section about the Sylow theorems. I have already read hints saying to use induction and that $p$-groups always have non-trivial centres, but I'm still confused. ...
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### References on the theory of $2$-groups.

Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to ...
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### If $H$ is a proper subgroup of a $p$-group $G$, then $H$ is proper in $N_G(H)$.

Let $H$ be a proper subgroup of $p$-group $G$. Show that the normalizer of $H$ in $G$, denoted $N_G(H)$, is strictly larger than $H$, and that $H$ is contained in a normal subgroup of index $p$. Here'...
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### Classify groups of order 27

Let $|G|=27$. Prove that all subgroups of index $3$ are normal. Classify all groups of order $27$. I can do the first one, but the classification is overwhelming. I don't even know where to start. ...
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### Conjugacy classes of a $p$-group

This is a problem from Preliminary Exam - Spring 1984, UC Berkeley For a $p$-group of order $p^4$, assume the center of $G$ has order $p^2$. Determine the number of conjugacy classes of $G$. ...
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### The $i$-th center $Z_{i}(G)$

Let $H$ be a normal subgroup of a $p$-group $G$, $H$ is of order $p^i$. Prove that $H$ is contained in the $i$-th center $Z_{i}(G)$. Recall that we define $Z_{0}(G)=1$, and for $i>0$, $Z_{i}$ is ...
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### Characterizations of the $p$-Prüfer group

I'm an undergrad student fairly keen on algebra. Over the different algebra courses I've taken, I've often encountered the so-called $p$-Prüfer group on exercises but somehow never got around to them. ...
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### A question on $p$-groups, and order of its commutator subgroup.

$\textbf{QUESTION-}$ Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$. If $P=Z(P)$ it is true. Now let $n > 1$, then If I see $P$ as a nilpotent group and ...
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Given: Let $G$ be a group, and let $\mathcal{S}$ be the set of subgroups of $G$. For $g\in G$ and $H\in S$, let $g\cdot H=gHg^{-1}$ Question: Deduce that if $G$ is a finite $p$-group, for some prime $... 1answer 165 views ### Bound on the number of p-groups for fixed exponent It's well-known that for each prime number$p$there are exactly two groups of order$p^2$, five of order$p^3$, and fifteen of order$p^4$(at least when$p>3$). I know that the classification of ... 1answer 155 views ### Groups of order$64$with abelian group of automorphism G. A. Miller in 1913 constructed the first example of a non-abelian group of order$64$with abelian group of automorphisms. It is the group$$G=(C_8\rtimes C_4)\rtimes C_2=\langle x,y,z\colon x^8, y^... 1answer 187 views ### An example of a simple infinite$2$-group Is there an example of a simple infinite$2$-group? Informations If a$2$-group is Artinian I know that it also locally finite, so the simple$2$-group cannot be Artinian. Take the subgroup ... 1answer 109 views ### Classify$p$-groups in which all groups of the same order are isomorphic The answer to “Are two subgroups of a finite$p$-group$G$, of the same order, isomorphic?” is definitely no. Such groups are very rare. How rare? Can you classify all finite$p$-groups$G$such that ... 1answer 251 views ### Frobenius kernel is regular normal elementary abelian p-subgroup? I'm attempting Exercise 3.4.6 in Dixon & Mortimer's book on Permutation Groups: Let$G$be a finite primitive permutation group with abelian point stabilisers. Show that$G$has a regular ... 2answers 145 views ### There is no core free subgroup of order$p^2$in a group of order$p^4$By the classification of group of order$p^4$($p$odd prime) from Burnside's book it seems to me that there is no core free subgroup of order$p^2$in a group of order$p^4$. If I am not wrong there ... 1answer 420 views ### An infinite$p$-group may not be nilpotent It is well-known fact that every finite$p$-group$G$is nilpotent. I am asking to have a counter example when$G$is infinite$p$-group. Thanks. 2answers 912 views ### Is this Galois theory proof of Fundamental Theorem of Algebra correct? I am studying Galois theory through Lang's Algebra and Dummit-Foote's Abstract Algebra. While studying the Fundamental Theorem of Algebra's proofs from both books I spent a lot of time to understand ... 1answer 103 views ### Definition of$\Omega$-group and$\Omega$-composition series What are the definitions of$\Omega$-group and$\Omega$-composition series? No luck searching on the internet.. 1answer 275 views ### Does every$p$-group of odd order admit fixed point free automorphisms? Does every$p$-group of odd order admit fixed point free automorphisms? equivalently, Given an odd order$p$-group$P$, is there a group$C$such that we can form a Frobenius group$P\rtimes C$? ... 1answer 83 views ### Are there asymptotically more nonabelian groups of order$p^k$than there are abelian groups of order$\leq p^k$? Let$\alpha(n)$denote the number of isomorphism classes of abelian groups of order$n$and$\alpha^\prime(n)=\sum_{m=1}^n\alpha(m)$. Similarly, define$f(p^k)$to be the number of isomorphism ... 1answer 347 views ### Why are there so many groups (up to isomorphism) of order$p^n$for$n>2$, especially when compared to groups of similar sized order? While bounds on the number of isomorphism classes of groups of order$p^n$where$p$is prime have been known for quite a while (such as the work of Higman$^{[1]}$and Sims$^{[2]}$) which give us the ... 0answers 117 views ### Abelian groups whose automorphism group is a$p$group$\def\Aut{\operatorname{Aut}}$Let$G$be a finite abelian group such that$\Aut(G)$is an$p$group ,that is,$|\Aut(G)|=p^n$. Then can we determine the cyclic decomposition of$G$or at least the ... 0answers 301 views ### Groups of order$p^5$I am reading a paper "A Determination of order$p^5$" by H A Bender ($p$is an odd prime). He divides the classification in two classes, one which contains an abelian subgroup of order$p^4$and other ... 5answers 2k views ### Any group of order$2p$has a subgroup of order$p~$($p$being a prime) I want to show without using Sylow theorem that Any group of order$2p$has a subgroup of order$p~$($p$being a prime) My attempt: Since$|G|=2p,$even$\exists~a\neq e\in G$such that$a^{-1}=a.$... 1answer 1k views ### Difference between definitions of$p$-subgroup and Sylow$p$-subgroup I'm reading Abstract algebra by Dummit and Foote and the following definitions are made:$1$. A group of order$p^{\alpha}$for some$\alpha\geq1$is called a$p$-group. Subgroups of$G$which are$...
Suppose $P$ is a finite abelian $2$-group and $U$,$V$ are characteristic subgroups of $P$ such that $|V:U|=2$. Does it follow that $P$ has a characteristic subgroup of order $2$?
$G$ is a p-group and $S$ is a set that $G$ acts on. p does not divide $|S|$. Why is there at least one element $a\in S$ such that $|O(a)|=1$, or in other words, $G_a=G$? I tried to ask this question ...