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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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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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110 views

Reference request: groups of order $p^4$.

I am looking for a textbook or a paper which include the classification of groups of order $p^4$ ($p$ is prime) using generators and relations. In particular I like to understand which group $G$ "...
6
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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 ...
6
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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 ...
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189 views

Is there an analogue of outer Space to study outer automorphisms of free pro-$p$ groups?

I would like to know if there is an analogue of Culler & Vogtmann's outer space to study outer automorphisms of free pro-$p$ groups. Perhaps an initial guess of such a space would be a moduli ...
4
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65 views

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 ...
4
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392 views

Classification of non abelian groups of order $p^3$.

This is not a duplicate of this post, neither of this: they don't give an explicit description of these groups, but only some of their properties. Using GAP to find all the non-abelian groups of ...
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261 views

Classify all groups of order $p^2q^2$ up to isomorphism

Let $p,q \in \mathbb{N}$ be prime numbers with the properties $2 < p < q$ and $q - 1 , q + 1 \notin \left\langle p \right\rangle$ Classify all groups of the order $p^2q^2$ up to isomorphism. ...
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57 views

Intersection of subgroups with normal series of prime index

Let $G$ be a group, and $H$ a subgroup such that for a prime $p$, there is a normal series $$ H = H^0 \triangleleft H^1 \triangleleft \cdots \triangleleft H^n = G $$ between $H$ and $G$ such that ...
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62 views

Tensor product with Prüfer $p$-group

Consider the Prüfer $p$-group $$\mathbb{Z}/p^{\infty} = \varinjlim\limits_n \mathbb{Z}/p^n \mathbb{Z}.$$ Let $A$ be any abelian group. Then $A \otimes \mathbb{Z}/p^{\infty}$ is a divisible $p$-torsion ...
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136 views

About Special and Extra-special $p$-groups

A $p$-group $G$ is said to be special $p$-group if $Z(G)=[G,G]=$ elementary abelian. A $p$-group $G$ is said to be extra-special if $Z(G)=[G,G]=$ elementary abelian of order $p$. The questions I ...
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70 views

Homomorphisms from a $p$-group to $\mathbb{F}_p$

I'm doing a problem on group cohomology and have reduced it to the following: if $P$ is a $p$-group then $\textrm{Hom}(P,\mathbb{F}_p) \simeq P/\Phi(P)$ where $\Phi(P)$ is the Frattini subgroup of $P$....
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402 views

Unitriangular group $UT_n(\Bbb Z)$ is nilpotent with class $n$

The unitriangular group $UT_n(\Bbb Z)$ is the group of all $n \times n$ invertible triangular matrices with the identity on each entry of the main diagonal, and integer entries everywhere else in the ...
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84 views

If a certain group action fixes every element $x$ such that $x^4=1$, then the action is trivial

This is a question from chapter $4D$ of Isaacs' Finite Group Theory. Let $A$ act via automorphisms on $G$, where $G$ is a $2$-group and $A$ has odd order. Show that if $A$ fixes every element $x$ in ...
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391 views

Properties of Finite and Infinite $p$-Groups

By a $p$-group, we mean a group in which every element has order a power of $p$. It is well known that finite $p$-group has non-trivial center. But, an infinite $p$-group may have trivial center. ...
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258 views

Finitely generated infinite $p$-group with a unique subgroup of order $p$

I'd like to ask can we characterize the structure of finitely generated infinite $p$-group which has a unique subgroup of order $p$? Can we say that these group are residually nilpotent? Any ...
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38 views

On $n$th class-preserving automorphism of finite $p$-group

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called an $n$th class-preserving if for each $x\in G$, there exists an element $g_x\in \gamma_n(G)$ ...
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33 views

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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69 views

Some conditions on a finite non-abelian $2$-group

Let $G$ be a finite non-abelian $2$-group, $\nu(G)$ denotes the number of conjugacy classes of non-normal subgroups of $G$ and $G^{\prime}$ denotes the derived subgroup of $G$. If $|G^{\prime}|=8$ ...
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69 views

cyclic subgroups of a $p$-group

Let $G$ be a finite non-Dedekind $p$-group and $\nu^*(G)$ denote the number of conjugacy classes of non-normal cyclic subgroups of $G$. Does there exist a normal second maximal subgroup $S$ of $...
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92 views

Regarding the nilpotency class of finite p-groups

So I'm aware that for a $p$-group $G$ of order $p^{n}$, say, that $G$ must have a nilpotency class between 1 and $n - 1$ for $n\geq 2$. My question is why can $G$ not have nilpotency class $n$? Take ...
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82 views

Is there any English version of p transfer theorem of Wielandt?

I want to read a proof of the following theorem of Wielandt, Theorem: Let $G$ be a finite group and let $P\in Syl_p(G)$ such that $P$ is regular. Then $N_G(P)$ controls $p$-transfer. I found a proof ...
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152 views

A finite non-abelian $p$-group contains at least an element of order $p^2$

Let $G$ be a finite non-abelian $p$-group such that $G$ contains at least an element of order $p^2$ and for every nontrivial normal subgroup $N$, $G/N$ has not any elements of order $p^2$ and $G/Z(G)$ ...
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145 views

p-Group satisfying the minimal condition on abelian subgroups

Are there examples of $p$-groups satisfying the minimal condition on abelian subgroups but do not satisfy the minimal condition? Obviously such a group cannot be locally finite.
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73 views

Factor group of a $p$-group

I asked this question a few days ago in MO but got no answer, so I try here. Any hint will be appreciated. Let $$M(p^3)=\langle a, b\mid a^{p^2}=b^p=1, a^b=a^{p+1}\rangle$$ and $$G=\langle a, b\...
2
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0answers
61 views

Index of a maximal subgroup among normal abelian subgroups

Let $P$ be a $p$-group and $A$ maximal among abelian normal subgroups of $P$. Show that: 1) $A=C_P(A)$. 2) $|P:A|\mid (|A|-1)!$. 1) If $A$ is an abelian normal subgrup of a certain group $G$...
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150 views

p-group of class 2 with cyclic commutator subgroup

I was reading a paper "ON THE EXISTENCE OF NONINNER AUTOMORPHISMS OF ORDER TWO IN FINITE 2-GROUPS" you can get it from here. In Introduction page 2 He said "It is worth mentioning here that we need ...
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144 views

An example where $Z(Z_G(A))$ is not a subset of right Engel elements in a finite $p$-group

Find a counter example to the following statement: Let $G$ be a finite $p$-group such that $G/Z(G)$ has exponent $p$. Let $A$ be a normal abelian maximal subgroup of $G$, and $Z_G(A)$ be the ...
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370 views

Lift a group action from a quotient

Let $p$ be a rational prime and $H$ be a finite cyclic group of prime order $l$ prime to $p$, i.e. $(l,p) = 1$. Let $G$ be a finite abelian group of $p$-power order. If I can write an (abelian) group ...
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A question to the ascending central chain in modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $char(K)=p$. It is well-known that the group $1+rad(KG)$ is a p-group containing $G$. Let us focus on the sequence $G\cdot ...
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61 views

Intersection of all maximal subgroups of a finite group

Let $G$ be a group. Define $$F=\left\lbrace x\in G\;:\; \text{if } \left\langle S,x\right\rangle=G\text{ then }\left\langle S\right\rangle=G\right\rbrace.$$ I have that $F$ is a normal subgroup of $G$....
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Why is the $p-$multiplicator of a $p-$group is an elementary abelian $p -$group?

Let $G$ be a $p-$group that have the finite presentation $F/R$($F$ is a free group of rank $d$); The $p-$multiplicator of $G$ is defined by $$ G^* = R/[F,R]R^p $$ Why $G^*$ in an elementary abelian $p ...
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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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35 views

Upper central series, semidirect products and a lifting property (argument verification)

Let $G$ be a profinite group, and define the lifting property: $(*_p)$ For every extension $1 \to P \to E \to W \to 1$ where $E$ is finite and $P$ is a $p$-group and for every surjective morphism $f\...
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36 views

Showing $d(\frac{Z_2(G)}{Z(G)})=d(G)d(Z(G))$ if and only if $d(G)=2$.

I am tring to proves following lemma Let $G$ be a finite p-group of coclass 3 and nilpotency class greater than 3 then $d(\frac{Z_2(G)}{Z(G)})=d(G)d(Z(G))$ if and only if $d(G)=2$ correct me ...
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18 views

Central quotient of $p$-groups

Suppose $P$ is a finite $p$-group with center $Z(P)$ of order $p$. What kind of groups can appear as the central quotient $P/Z(P)$? For example, the quotient is in particular a so-called capable ...
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19 views

Determine possible $p$-groups from center and quotient

Consider the following situation: I have given a finite $p$-group $P$ (in the case I am interested in $p = 2$) with cyclic center $Z(P)$ and I also know the structure of the quotient $P/Z(P)$ (which ...
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29 views

Subgroup of index p in an infinite p-group?

Does an abelian infinite $p$-group always contain a subgroup of index $p$ ? Thanks.
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84 views

Does $p$ always divide $[N_G(H) : H]$ when $G$ is a $p$-group?

On this question, $G$ is a finite $p$-group and $H$ is a proper subgroup of $G$. The accepted answer says Note that p divides $|N_G[H]/H|$ so $N_G[H]/H$ has a subgroup of order $p$. But what ...
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129 views

Subextension of a field with Galois series of subextensions of prime degree

Let $p$ be a prime number, and $E/F$ be a field extension. Suppose $E/F$ has a finite series of subfields $$ F = E_0 < E_1 < \cdots < E_n = E $$ with $E_i / E_{i-1}$ Galois of degree $p$ ...
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147 views

Normal p-subgroup and maximal subgroup

Let $G$ be a finite group and $p$ a prime s.t. $p\big||G|$, and let $P$ be a normal p-subgroup of $G$, with $|P|=p^m$. I want to prove the following: If $M$ is a maximal subgroup of $G$, then $P\...
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45 views

Ulm Invariants of a reduced abelian primary group

Let $G$ be a reduced abelian primary group of lenght $\lambda$, and let $\alpha$ and $\beta$ with $\beta$ a limit ordinal and $\alpha < \beta \leq \lambda$. Show that the Ulm Invariants of $G_{\...
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0answers
57 views

Lengths of $p$-central $p$-series

The lower $p$-central $p$-series of a finite $p$-group $G$ is defined as follows: $$G=P_1(G)>P_2(G)>P_3(G)>\cdots P_m(G)=1,$$ where $P_i(G)$ is the smallest normal subgroup of $G$ contained ...
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19 views

$p$-groups as specific central quotients

There are $p$-groups such that they can not be isomorphic to $G/Z(G)$ for any group $G$. Perhaps such thing may not arises if we replace $Z(G)$ by a term of lower central series. Then my question is ...
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0answers
54 views

Center of a subgroup of a certain $2$-group

Let $G$ be a finite $2$-group of class 2 (that is $G^{\prime}\leq Z(G)$) and $Z(G)$ be cyclic. We could verify that $G^{\prime}=\langle [a,b]\rangle$ for some $a, b \in G$. If $H=\langle a,b\rangle$,...
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0answers
260 views

Prove a function to be an automorphism of a $p$-group $G$.

Let $M$ be a maximal subgroup of a $p$-group $G$. For fixed $g\in G\backslash M$ and $z\in Z(G)\cap M$ of order $p$, the map \begin{align*} \alpha : G&\longrightarrow G\\ mg^i &\...
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168 views

Characteristic subgroups of non-abelian $p$-group

It is a difficult problem to determine all the characteristic subgroups of a non-abelian $p$-group. But, then, I would seek for as many characteristic subgroups as we can, for small order $p$-groups. ...
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0answers
31 views

Homomorphisms $G\to\text{Aut}(G)$ for a $p$-group

I am interested in studying automorphisms of a $p$-group, or at least the highest power of $p$ dividing the order of the automorphism group, and feel like studying homomorphism $G\to\text{Aut}(G)$ ...
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0answers
74 views

On centers of infinite $p$-groups and nilpotent groups

If $G$ is a finite $p$-group, then its center is non-trivial, which forces that $G$ must be nilpotent. Consider infinite $p$-groups, i.e. infinite groups in which order of every element is some power ...
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175 views

example of p-group with infinite Frattini subgroup

Is there an infinite $p$-group $G$ with infinite Frattini subgroup $\Phi(G)\not = G$? In "Subgroups of Teichmuller Modular Groups" there is an example but I don't get it because I don't know much ...