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.
620
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Quotient of a minimal non abelian $p$-group by its center
I am trying to solve the following question:
Let $P$ be a nonabelian finite $p$-group in which every proper subgroup is abelian. Show that $P/Z(P) \simeq C_p \times C_p .$
Since $P$ is non-abelian I ...
- 67
-1
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1
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47
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Center of a nonabelian p group can be found by taking the intersections of centers of two suitable maximal subgroups
Let $P$
be a nonabelian $p$
group. Prove that there exist maximal subgroups $M$ and $N$ of $P$ such that
$$Z(P)=Z(M) \cap Z(N).$$
I found that $Z(P) \subseteq Z(M) \cap Z(N).$ I have no idea for the ...
- 67
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0
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96
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Conjugacy classes of a non-abelian group of order $p^3$
Let $G$ be a nonabelian group of order $p^3$, where $p$ is a prime. It's well known that $|Z(G)|=p$. The noncentral elements have all centralizer of order $p^2$, because $Z(G)<C_G(x)<G$ for ...
- 2,374
3
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2
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72
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Prove/disprove: Let $G$ s.t $|G| = p^3 \implies |G'| \leq p$. [duplicate]
Let $G$ s.t $|G| = p^3$ for some prime $p$. Prove or disprove: $|G'| \leq p$.
I couldn't think of any counter-examples so I started proving this and I'm stuck unfortunately and was hoping to seek ...
- 411
1
vote
0
answers
22
views
2-groups with abelianization of type (2,2)
On Olga Taussky's article "A Remark on the class field tower" there is a note on the last page where she states that P. Hall pointed out to her that exists 3 different groups of order $2^k$ ...
0
votes
1
answer
57
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Cauchy's theorem for groups use unclear [duplicate]
Im reading about the Burnside's $p^aq^b$ theorem proof, and for the case when $b=0$ it uses that, since the order of the group is a power of a prime, by an elementary result of group theory we have ...
0
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1
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46
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Proving G in a p-group using the centralizer of its elements
I came across this question in a past exam in an Algebraic structures course. It's a first course in group theory, and I had some difficulty solving this problem.
Let $ p \in \mathbb{N}$ be a prime ...
- 19
1
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0
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91
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Normalizer of subgroup of a finite $p$-group. [duplicate]
I have been struggling with the following exercise for a while now:
Let $p$ be a prime number and $G$ be a finite $p$-group. Show for any real subgroup $H$ of $G$ that $H$ is a real subgroup of the ...
- 11
2
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1
answer
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In a nonabelian group of order $p^4$, a maximal normal and abelian subgroup of $G$ is of order $p^3$.
Let $p$ be a prime and $G$ be a nonabelian group of order $p^4$. Let $H$ be a subgroup of $G$ maximal with respect to $H$ being normal in $G$ and $H$ being abelian. We have to show that $|H|=p^3$.
I ...
- 2,413
2
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0
answers
32
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Explicit construction of an outer isomorphism in a $p$-group.
Recently, I stumbled upon a theorem by Wolfgang Gaschütz (see below) that every non-trivial $p$-group which is non-trivial has an outer automorphism. However, the proof uses cohomology theory and as ...
- 197
3
votes
1
answer
70
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If $G/G’ $ is cyclic, prove by induction that $G/Z(G)$ is cyclic if $G$ is $p$ group. [duplicate]
Let $G$ be a $p$ group. If $G/G’$ is cyclic, prove $G/Z(G)$ is cyclic by induction.
I tried playing with the base cases : if $|G|=p $ or $p^2$, $G $ is abelian, so $G/Z(G)=G/G =\{G\}$ which is ...
- 1,005
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0
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45
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The Second Cohomology group with Coefficient in cyclic group of order 3.
I am computing the second cohomology group $H^2(G,\mathbb{Z_3})$ of $G=$SmallGroup(81,9). Its presentation is given by
$G=\langle x,y,z\mid x^3=y^9=z^3=[y,z]=1,[y,x]=z,[z,x]=zy^3z^{-1}\rangle$.
I ...
- 51
1
vote
1
answer
63
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Let $|G|=p^a$. Prove that if $|N|=p^{a-1}\Rightarrow N\lhd G$
Let $|G|=p^a$. Prove that if $|N|=p^{a-1}\Rightarrow N\lhd G$
If $a=1$ this is straightforward because $G$ is abelian and every group is normal, in particular $N$. Let us take $a>1$.
First try: If ...
- 1,005
1
vote
0
answers
52
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Showing that the direct sum is an infinite $p$-group which is not nilpotent.
I am trying to solve problem 5.45 from Rotman's book:
For each $n\geq 1$ , let $G_n$ be a finite $p$-group of class $n$. Define $H$ to be the group of all sequences $(g_1, g_2, ... )$, with $g_n\in ...
- 69
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0
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46
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Is this a non-degenerate bilinear form on a quotient of a special p-group?
My question: Is the bilinear form $B$ defined below non-degenerate in general?
Let $P$ be a non-abelian special $p$-group $\left(Z(P) = [P,P] = \Phi(P)\right)$ with $Z = Z(P) \cong C_{p}^{n}$ ($n \geq ...
- 11
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Is it possible to prove $(G:H)\equiv (N:H) \pmod{p}$ without using Orbit Stabilizer Theorem or Sylow Theorem?
$G$ is a finite group with order $|G|=p^am$, where $p$ is prime and $p\nmid m$. $H$ is a p-subgroup of $G$, so $|H|=p^i$, where $1\le i\le a$. Define: $N=\{g\in G| gHg^{-1}=H\}$. Prove: $(G:H)\equiv (...
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1
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63
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A quotient group of a $p$-group is a $p$-group [closed]
I'm trying to prove that given a $p$-group $G$ and any normal subgroup $A$ of $G$, the quotient group $G/A$ is also a $p$-group.
If $G$ is finite, then $A$ is finite too and the size of $G/A$ is a ...
- 11
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0
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61
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Structure Description of p-groups of nilpotency class 2, where p is an odd prime.
Suppose $G=\langle a,b \mid [a,b]^{p^\gamma}=[a,b,a]=[a,b,b]=1, a^{p^{\alpha}}=[a,b]^{p^\rho}, b^{p^{\beta}}=[a,b]^{p^\sigma}\rangle$, where $\alpha>\beta\geq \gamma\geq1$ and $0\leq\sigma<\rho&...
- 51
1
vote
1
answer
164
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Non-$p$-torsion difference of $p$-torsion elements in a $p$-group
I have a relatively simple question. For $p$ odd, does there exist a non-abelian $p$-group such that for every pair of non-commuting $p$-torsion elements $g$,$h$ their difference $g^{-1}h$ is not $p$-...
- 536
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1
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38
views
Is there a natural number $k$ such that for every prime $p$ there is a non-Abelian group of order $p^k$?
Is there a natural number $k$ such that for every prime $p$ there is a non-Abelian group of order $p^k$?
Update: $k=3$ should work; there is a nontrivial semidirect product $(\mathbb{Z}/p^2\mathbb{Z}) ...
- 2,391
4
votes
1
answer
77
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Minimal presentation of non-abelian group of order $p^3$ and exponent $p^2$
Let $p$ be an odd prime and let $H$ be the non-abelian $p$-group of order $p^3$ and exponent $p^2$. A presentation of $H$ is given by
$$H=\langle a,b \mid a^{p^2}=1,b^p=1,[a,b]=a^p \rangle. $$
How do ...
- 1,026
2
votes
1
answer
43
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Height of $px$ in a $p$-group.
Let $G$ be an abelian $p$-group, meaning that every element has order $p^n$ for some $n$. We can define, inductively, $G_0 = G$, $G_{\alpha + 1} = pG_{\alpha}$, and $G_{\alpha}$ for limit $\alpha$ is ...
- 388
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votes
2
answers
91
views
Which $p$- groups satisfy $G^p\gamma_2(G)=G^p\gamma_3(G)$?
Let $G$ be a finite $p$-group. Let $\gamma_n(G)$ denote the lower central series of $G$. In particular we have $\gamma_2(G)=[G,G]$ and $\gamma_3(G)=[[G,G],G]$. Let $G^p$ be the subgroup generated by $...
- 1,026
3
votes
1
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72
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If $G$ is a finite $p$-group, then the socle of $G$ is contained in the centre $Z(G)$
This is an exercise in Permutation Group by D.Dixon and Brian Mortimer, from page112, exercise 4.3.3.
Recall: The socle of $G$ is the subgroup generated by the set of all minimal normal subgroups of $...
- 124
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0
answers
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Show that this finite $p$-group is isomorphic to a product of two cyclic groups of the same order
Let $G$ be an abelian group of $p^{2n}$. Suppose for every $1 \le r \le n$, the subgroup $G_r=\{g \in G:g^{p^r}=e\}$ has order $p^{2r}$. Show that $G \cong \mathbb{Z}/p^n\mathbb{Z} \times \mathbb{Z}/p^...
- 258
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2
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139
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Prove a simple module must be "trivial"
Let $P$ be a finite $p$-group, i.e. $|P|=p^n$ for some prime $p$. Let $F$ be a finite field of $p$ elements. Show that every simple $FP$-module trivial, where $FP$ is a group ring (which is actually a ...
- 2,195
1
vote
1
answer
41
views
Image of augmentation ideal of a group ring under ring homomorphism.
Let $G,H$ be two finite $p$-groups and $f:\mathbb{F}_p[G]\to \mathbb{F}_p[H]$ a ring homomorphism of mod $p$ group rings.
Is it always true that $f(I_G)\subset I_H$, where $I_G,I_H$ are the respective ...
- 1,026
2
votes
0
answers
33
views
Prove that $K’’=1$ for a particular $p$-group $K$
Let $S$ be a $p$-group, $N$ a normal subgroup of $S$ contained in $Z(S)\cap S’ $.
Consider $K$, another normal subgroup of $S$ with these two properties:
$K/K’$ is elementary abelian
$[K,K,K]=N$
How ...
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votes
1
answer
88
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How to show that $|G|=d_1^2+\cdots +d_s^2$ and $G$ is abelian?
In Representation Theory of Finite Groups: An Introductory Approach by Benjamin Steinberg, there is a corollary in which I feel confused.
Corollary 6.2.6. Let p be a prime and let $G$ be a group of ...
- 91
2
votes
1
answer
144
views
What are all the finite groups where every nontrivial element has order $3?$
I know that the powers of any group where all elements have order $3$ also has this property. I also know that the Heisenberg group has this property and that all finitely generated such groups are ...
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Group of order $2p$ where $p$ is an odd prime is isomorphic to either integers mod $2p$ or dihedral group of order $2p$.
$\newcommand{\Z}{\mathbb{Z}}$
Proposition: Let $p$ be an odd prime and $G$ a group of order $2p$. Then either $G \cong \Z/2p\Z$ or $G \cong D_{2p}$.
Attempt:
Well $G$ can be either cyclic or non ...
- 2,377
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0
answers
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Any group of order $p^2$ is abelian, where $p$ is prime; $G/Z(G)$ cyclic implies $G$ is cyclic. [duplicate]
I've taken an approach and it really feels like it should be getting me there, but I can't seem to close it off. Any hints?
Attempt: If $G$ is a group of order $p^2$ then as a $p$-group $Z(G)$ is non-...
- 2,377
3
votes
2
answers
127
views
Showing that $|X| \equiv |\text{Fix}(G)|\pmod{p}$.
$\newcommand{\fix}{\text{Fix}}$
$\newcommand{\stab}{\text{stab}}$
$\newcommand{\orb}{\text{Orb}}$
Definition: Let $G$ act on a set $X$. We define $\fix(G) =\{x \in X : g \ast x = x, \forall g \in G\}$...
- 2,377
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vote
0
answers
28
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Question on self-normalising $p$-subgroups of $p$-groups [duplicate]
Given a finite $p$-group $P$, we call a proper subgroup $Q<P$ centric if $C_P(Q)=Z(Q)$.
The question is: can a centric subgroup $Q$ be self-normalising, which means that $N_P(Q)=Q$?
I do not see ...
- 329
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56
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Groups of order $p^3$, $p$ prime [duplicate]
I know that there are the abelian groups $(\mathbb{Z}/p)^3$, $\mathbb{Z}/p^2 \times \mathbb{Z}/p$ and $\mathbb{Z}/p^3$, and the non-abelian Heisenberg group of matrices $
\left[ {\begin{array}{cc}
...
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2
answers
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views
Let $G$ be a $p$-group and let $H$ be a proper subgroup of $G$. Show that $H$ is a proper subgroup of $N_G(H)$
Let $G$ be a $p$-group and let $H < G$. Show that $H < N_G(H)$.
If $H \trianglelefteq G$, then it should be clear that $H<N_G(H)$. So we suppose, $H$ isn't normal. So we let $H$ act on the ...
- 1,700
2
votes
1
answer
64
views
Finite subgroup containing all finite subgroups of infinite group
The proof of the following
Theorem In an infinite 2-group every finite subgroup is properly contained in its normalizer.
begins with
Let $F$ be a finite subgroup of an infinite 2-group $G$ and ...
- 128
3
votes
1
answer
45
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An alternative notion of nilpotency class for $p$-groups
Let $G$ be a finite $p$-group for some prime $p$, and let $\rho: G \to \text{Aut}(A)$ a faithful representation of $G$ for some finite abelian $p$-group $A$ (which exists because, for example, we may ...
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4
votes
1
answer
75
views
Are metacyclic $p$-groups semidirect products?
A group $G$ is called metacyclic if there is cyclic $N\unlhd G$ such that $G/N$ is cyclic as well. If $G$ is a metacyclic $p$-group, I know that there is a presentation
$$G\cong\langle x,y\mid\, x^{p^...
- 175
1
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1
answer
61
views
$U^s = (U^p)^s$ and there exists a normal series with factor group a p-group
Let $U$ a finite group.
We can define $U^p$ the smallest normal subgroup of $U$ st $U/U^p$ is a $p$-group and analogously $U^s$ the smallest normal subgroup of $U$ st $U/U^s$ is solvable (both thanks ...
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0
answers
74
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A question on a 2-group with an elementary Abelian maximal subgroup
Let $G$ be 2-group of order $2^{n+1}$($n\geqslant2$) which has a maximal subgroup $N\cong\mathbb{Z}_{2}^{n}$. It is straightforward to check that if $G$ is an Abelian group, then $G$ is isomorphic to $...
- 351
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votes
3
answers
114
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Smallest normal subgroup such that the factor group is a $p$-group
I am working with an Andreas Dress's article, and he says
for a group $U$ (finite), the subgroup $U^p$ is the (well defined !) smallest normal subgroup of $U$ with $U/U^p$ a $p$-group.
I think that ...
- 231
1
vote
1
answer
83
views
Let $G$ be a group order order $p^3$. Show for any $g,h \in G$, we have $g^p h = hg^p$
Let $p$ be a prime number. Let $G$ be a group order order $p^3$. Show for any $g,h \in G$, we have $g^p h = hg^p$
If $|g|=1$, then we're done.
If $|g|=p$, then we're done.
If $|g|=p^3$, then group is ...
- 1,516
3
votes
1
answer
104
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Prove that every group of order $p^m$ with $p$ a prime number can be generated by $m$ elements
Prove that every group of order $p^m$ can be generated by $m$ elements, where $p$ is a prime number.
I am thinking about induction.
Base case: if $m=1$, then $|G|=p$. So $G$ is cyclic. Hence, it's ...
- 1,516
2
votes
1
answer
113
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Classification of nilpotent groups via $p$-groups
First some background: Some time ago I learned that since any finite abelian group is a direct product of (finite) cyclic groups, the finite cyclic groups ($\mathbb{Z}_n$) are the key to understanding ...
- 3,013
3
votes
0
answers
81
views
Let $G$ be finite & nonsimple such that all normal subgroups of $G$ are $p$-groups for a given prime $p$. Is it true that $G$ is also a $p$-group?
Let $G$ be a finite non-simple group such that all normal subgroups of $G$ are $p$-groups for a given prime $p$. Suppose also that the $p$-Sylow subgroups of $G$ are not normal. Is it true that $G$ is ...
- 307
3
votes
0
answers
66
views
How we illustrate the Prüfer-$3$ group elements on the unit circle?
I am trying to illustrate the Prüfer-$3$ group elements on the unit circle and I am not sure, if my approach is correct. It should be noted that I am mainly interested in the odd integers.
Let us ...
- 1,574
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votes
1
answer
62
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Prove if $G$ is a finite nonabelian $p$-group, then $p^2\mid |{\rm Aut}(G)|$. [duplicate]
Prove if $G$ is a finite nonabelian $p$-group, then $p^2\mid|{\rm Aut}(G)|$.
Suppose $|G|=p^m, m\in \mathbb{N}$.
A fact I know about $p$-groups:
Since $G$ is a $p$-group, $\forall i\leq m \space \...
- 2,292
3
votes
1
answer
93
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Show that $q\mid |N_G(H\cap K)|$ where $H,K\in\mbox{Syl}_p(G)$ with $|G| = p^nq$.
Let $G$ be a group of order $p^nq$ where $p,q$ are distinct prime numbers and $n$ is positive integer. Suppose $G$ has at least two distinct Sylow $p$-subgroup $H'$ and $K'$ such that $H'\cap K'\neq 1$...
- 4,682
1
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0
answers
46
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Given the background, how to show any map onto a cyclic group of order $p$ is of this specific form?
My question arised from an explanation to the proof of another problem. Here is the original problem, posted here:
For a finite elementary abelian group, the number of subgroups of A of order p ...
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