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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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Question about a possible application of Burnside's basis theorem

Let $G=\langle a_1,\dots , a_d\rangle$ be a $d$-generator $p$-group of order $p^n$ (i.e. $d$ is the minimal number of generators). Further let $N$ be a characteristic and elementary abelian subgroup ...
Aericura's user avatar
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1 answer
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How to check whether a finite $p$-group is regular in GAP?

I am trying to check whether a given $p$-group is a regular $p$-group in GAP. I am trying to use the command 'IsRegularPGroup(G)' for it. However I am getting 'Error, Variable: 'IsRegularPGroup' must ...
cryptomaniac's user avatar
0 votes
0 answers
41 views

Questions on a proof on $p$-constrained groups

Theorem: Let $G$ be a group and $p \in \pi(G)$. Furthermore, suppose that \begin{equation}\label{eq_p-constrained} C_{G/O_{p'}(G)}(O_p(G/{O_{p'}(G)})) \leq O_p(G/{O_{p'}(G)}). \end{equation} If $P$ ...
Stippinator's user avatar
1 vote
0 answers
37 views

Proof of Thompsons $A \times B$-lemma

(Auxiliary lemma) Let $G$ be a $\pi$-group and $a$ a $\pi'$-element acting on $G$. If $X$ is a subnormal subgroup of $G$ with $[a, X] = 1 = [a, C_G(X)]$, then $[a, G] = 1$. Hey guys, I am having a ...
Stippinator's user avatar
0 votes
1 answer
37 views

Structure of $p$-primary Abelian groups without Divisorial Elements [closed]

Let $A$ be a $p$-primary (in particular, torsion) Abelian group ($p$ prime). Assume that $A[p]=\{a \in A \vert pa=0\}$ is finite and that there are no nontrivial infinitely $p$-divisible elements, ie $...
user267839's user avatar
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1 vote
2 answers
91 views

Question regarding the properties of an automorphism group of a Sylow P subgroup

The context for this question has to do with proving: Groups of order $pq$ with $p < q$ have a normal subgroup of order $q$ and are cyclic iff $q$ is not congruent to $1$ mod $p$. I will leave out ...
froitmi's user avatar
  • 87
0 votes
1 answer
68 views

Prove that the center of a group is non-trivial [duplicate]

Consider the identity $|Z(G)|=|G|-\sum_{j=1}^m |C_{x_j}|$ where $C_1,...C_m$ are the conjugation classes in which $G\backslash Z(G)$ is partitioned into and $G$ is a non-abelian group with $p^m$ ...
Xaver Wallenstein's user avatar
1 vote
0 answers
41 views

Can we classify the structure of an infinite soluble $p$-group of bounded exponent?

By Baer-Pruffer Theorem, we know that, if $A$ is an abelian $p$-group of bounded exponent, then $A$ is a direct sum of cyclic $p$-groups. Now, assume that $A$ is an infinite soluble $p$-group of ...
Reza Fallah Moghaddam's user avatar
4 votes
1 answer
48 views

For a reduced abelian $p$-group $G$, does $P$ being finite imply $G$ is finite?

Let $G$ be a reduced abelian $p$-group, and let $P$ be the subgroup of elements of order $p$. If $P$ is finite, is $G$ necessarily also finite? Recall that a group $G$ is said to be reduced if it ...
rea_burn42's user avatar
2 votes
1 answer
46 views

No conjugacy class of size $p^{n-1}$ in group of order $p^n$ ($n \geq 2$)

Exercise. Let $p$ be prime and $G$ a group of order $p^n$ where $n \geq 2$. Show that there is no conjugacy class of size $p^{n-1}$. I've tried a few things, like applying the "$G$ mod center ...
azimut's user avatar
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0 votes
1 answer
77 views

Suppose primes $p,q\mid |G|$ with $p<q$. For $|x|=p$, does $x^G=(x^{-1})^G\iff p=2$?

The Question: Let $G$ be a group. Suppose primes $p,q\mid |G|$ with $p<q$. Let $x\in G$ such that $|x|=p$. Does $x^G=(x^{-1})^G$ if and only if $p=2$? Here $g^G$ is the conjugacy class of $g$ in $...
Shaun's user avatar
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2 votes
1 answer
104 views

The order of center of any non-abelian $p$-group of order $p^n$

Let $G$ be a non-abelian $p$-group of order $p^n$ with $n>3$. In this paper there is (just after Lemma 2.10) a statement which says $p^2 \leq |Z(G)|\leq p^{n-2}$. I know that the center of a $p$-...
Mahtab's user avatar
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1 vote
1 answer
104 views

Counting homomorphisms from $S_n$ to a $p$-group

There are plenty of exercises and questions counting homomorphisms between groups. However, the following has not been asked yet and I can not see any way to count the homomorphisms using the common ...
Chris's user avatar
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0 votes
1 answer
47 views

The number of direct sum of elementary abelian 2-groups

Let $G=(Z_2)^n$, I want to know the number of direct sum of $G$($G=H \oplus K$) or a fine upper bound. For $G=Z_2 \oplus Z_2$, I have calculated that all of its direct sum decomposition is as follows: ...
zeyu hao's user avatar
  • 347
0 votes
1 answer
32 views

MAGMA: How to efficiently do coercion of element of HomGrp to element of GrpAuto? [closed]

Suppose I have a finite $p$-group $G$ as GrpPC in MAGMA. The computation of the automorphism group $\mathrm{Aut}(G)$ takes a very long time. Suppose that I also ...
Aericura's user avatar
  • 291
4 votes
1 answer
23 views

Huppert, III.19.2: How to construct a homomorphism from a $p$-group into the center of a maximal subgroup?

Let $G$ be a finite $p$-group and $N$ a maximal subgroup (so $G/N$ has order $p$) such that $Z(N) \leq Z(G)$. III.19.2 in Huppert's Book "Endliche Gruppen I" says that there exists a non-...
Aericura's user avatar
  • 291
2 votes
1 answer
76 views

$p$-group with a cyclic subgroup

Throughout studying a paper about finite $p$-groups, I have the following question Let $G$ be a finite $p$-group with nilpotency class 3 and $\gamma_i(G)$ denote the i'th term of the lower central ...
shankfei's user avatar
  • 339
2 votes
0 answers
73 views

2-generated p-groups with generators of order p

Let $G=\langle a,b\rangle$ be a finite $p$-group such that $a^p=b^p=1$. Is there any result about the size of the set of $p$-elements $\Omega(G):=\{g\in G\mid g^p=1\}$? In particular, I'm interested ...
TommasoT's user avatar
  • 104
0 votes
1 answer
31 views

$p$-groups with nontrivial intersection of nonnormal subgroups

I study the following paper about finite $p$-groups, Finite groups in which the non-normal subgroups have nontrivial intersection, N. Blackburn, Journal of algebra, 3, 30-37 (1966). In this paper ...
shankfei's user avatar
  • 339
0 votes
1 answer
36 views

Show that the intersection of the distinct subgroups of a $p$-group of index $p$ is normal.

I've been struggling with this exercise (Exercise 8.2.28, Introduction to Abstract Algebra by Nicholson) for a few hours, and I haven't made much progress. Let $G$ be a group of order $p^n$ and let $...
iwjueph94rgytbhr's user avatar
0 votes
1 answer
74 views

About the two step centralizer

Let $G$ be a finite group. Define $\gamma_2(G)=[G,G]$ and $\gamma_{i+1}(G)=[\gamma_i(G),G]$ for all $i≥2$. Let $G$ be a finite $p$-group of order $p^n$ and maximal class. For each $i$ with $2≤i≤n−2$, ...
Fouad El's user avatar
  • 371
1 vote
0 answers
40 views

Subgroups and Quotient Groups of Infinite bounded Abelian p-groups

I am looking to a reference to a solution of Exercise 13 (page 91) of L. Fuchs, Infinite Abelian Groups, Vol. 1, Academic Press, 1970. There is no solution to this exercise in a collection Exercises ...
George's user avatar
  • 11
2 votes
2 answers
118 views

If $H, K \leq G$ implies $H \subseteq K$ or $K \subseteq H$, then $G$ is a (not necessarily finite) $p$-group.

Let $G$ be a group with the following property: for every $H, K \leq G$, either $H \subseteq K$ or $K \subseteq H$. Show that there exists a prime number $p$ such that the order of every element of $G$...
huh's user avatar
  • 464
3 votes
1 answer
54 views

The number of Hall $\pi$-subgroups of a $\pi$-separable group - Alexandre Turull article

This is an article which Alexandre Turull wrote. Lemma 2.1. states Lemma 2.1. Suppose $H$ is a finite group, acting on the finite group $F$, and assume that $|H|$ and $|F|$ are relatively prime. ...
math_survivor's user avatar
1 vote
1 answer
22 views

Sections of lower exponent $p$-series are elementary abelian

Let $$G = P_0(G) > P_1(G) > \dots > P_c(G) = 1$$ be a lower exponent $p$-series of a finite $p$-group $G$, i.e. $P_{i+1}(G) = [G, P_i(G)]P_i(G)^p$. In the $p$-quotient algorithm, one ...
Aericura's user avatar
  • 291
0 votes
1 answer
38 views

How are "types" defined in this module?

Note: This might end up being a question about a simple concept that I forgot about (I am very tired at the time of writing this, after all), so maybe try skipping to the bottom. I'm learning about ...
iwjueph94rgytbhr's user avatar
0 votes
2 answers
74 views

$G$ a non-cyclic group with $|G|=p^n$. Show that $G$ has at least $p + 3$ different subgroups.

Question: Let $p$ be a prime integer, $n \in \Bbb{N}$, $n \gt 1$, and $G$ a non-cyclic group with $|G|=p^n$. Show that $G$ has at least $p + 3$ different subgroups. I know that since $G$ is non-...
mathman's user avatar
  • 65
6 votes
1 answer
83 views

Why is a periodic subgroup of a residually p-group a p-group?

This might be a stupid question since I don't have much knowledge in group theory, but I've been struggling on this for a bit of time ; I am currently reading Wehrfritz's book on linear groups, and ...
NaCl's user avatar
  • 63
2 votes
1 answer
87 views

Is there a countable reduced $p$-group such that the corresponding Ulm-Kaplansky invariants are infinite or zero?

We know that countable Abelian groups are classified up to isomorphism by their Ulm-invariants. In the course of research, the following questions have been raised for me. If anyone has an answer in ...
Mahmood Behboodi's user avatar
2 votes
1 answer
149 views

Prove that every group $G$ with $p^n$ ($n\ge4$) elements and center with $p$ elements has an abelian subgroup of order $p^3$

I'm new in this forum so I hope I haven't made any mistake. I have to prove the above assertion. I've already proven that every such $G$ has an element $x$ whose centralizer has $p^{n-1}$ elements. I ...
Francesco Bradanini's user avatar
0 votes
0 answers
32 views

regular representation of a p group over field of characteristic p is indecomposible. [duplicate]

I know that the only irreducible representation of a p-group over field of characteristic p is the trivial representation. But I don't know how to prove this one?
Sagnik Dutta's user avatar
1 vote
0 answers
43 views

Exponent of a semi-direct product of nilpotent groups

It is well-known that in a semi-direct product of two groups $G = N\rtimes_\phi H$ the exponent of $G$ might be bigger than the lcm of the exponents of $N$ and $H$. It sometimes does not, I give an ...
Archistin Beedle's user avatar
1 vote
1 answer
73 views

Maximal abelian subgroups of an extraspecial group of order $2^{2m+1}$

I've found a proof of the structure of maximal abelian normal subgroups of an extraspecial group of order $2^{2m+1}$ in the book "Endlichen Gruppen I" by B. Huppert but there is a part of ...
Vicent Miralles's user avatar
2 votes
1 answer
47 views

If $G$ is a finite $p$-group do we have that $|[G',\gamma_i(G)]:[G',\gamma_{i+1}(G)]|\geq p$?

If $G$ is a finite $p$-group and the commutator subgroup $[G',\gamma_i(G)]\not=1$, can we ensure that the index $|[G',\gamma_i(G)]:[G',\gamma_{i+1}(G)]|\geq p$? Here $G'$ is the derived subgroup, and $...
Gillyweeds's user avatar
2 votes
2 answers
101 views

A property in finite $p$-groups

I study a paper about finite $p$-groups and I could not understand the following property which is used in it. Any comment or answer will be appreciated! Let $G$ be a finite non-abelian $p$-group and $...
shankfei's user avatar
  • 339
0 votes
0 answers
37 views

Give a non-trivial direct sum decomposition for the generalized Prufer $𝑝$-group $H_{\omega+1}$

We know that the generalized Prüfer $p$ -group $H_{\omega+1}$ is a group having the following generators and relations respectively $$X=\{a_0,a_1,a_2,\ldots\},~~~ \{pa_0=0, p^na_n=a_0,~ \text{for all}~...
Mahmood Behboodi's user avatar
2 votes
2 answers
117 views

If $H$ is a maximum subgroup of a finite group $G$ of index $p$ then $G$ is cyclic of order $p^n$

Consider the following statement: Let $G$ be a finite group, and $H\lneq G$ a proper subgroup that is maximum, meaning that every proper subgroup $H'\lneq G$ is contained in $H$. If $[G:H]=p$ with $p$...
Luigi Traino's user avatar
1 vote
1 answer
86 views

Proof of the Index Property for a Proper Subgroup in Finite $p$-Groups

Based on this question: proper subgroups of finite p-groups are properly contained in the normalizer. Let $G$ be a finite $p$-group and let $H$ be a proper subgroup. Then there exists a subgroup $H'$ ...
H.Y Duan's user avatar
  • 366
0 votes
1 answer
179 views

The number of double coset of p-group $G$

Following this question: Question about orbits of the left translation of group action Let $H$ be proper subgroups of the group $G$. For each $x \in G$ define the $H$ double coset of $x$ in $G$ to be ...
Hermi's user avatar
  • 1,514
0 votes
0 answers
75 views

The Number of conjugate classes of a non-abelian group of order $729$ [duplicate]

Howti identify Number of conjugate classes of a non-abelian group of order $729$. A group of order $p^3$ where $p$ is prime have $p^2+p-1$ number of conjugacy class.But my question is what is the ...
Nothing's user avatar
  • 11
1 vote
2 answers
126 views

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 ...
Q.E.D's user avatar
  • 77
-1 votes
1 answer
59 views

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 ...
Q.E.D's user avatar
  • 77
0 votes
0 answers
117 views

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 ...
Kan't's user avatar
  • 3,385
3 votes
2 answers
85 views

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 ...
MathStudent101's user avatar
2 votes
0 answers
29 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 $2^k$ ...
Vítor Machado's user avatar
0 votes
1 answer
64 views

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 ...
Guillermo García Sáez's user avatar
0 votes
1 answer
50 views

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 ...
Daniel's user avatar
  • 119
1 vote
0 answers
103 views

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 ...
martinr's user avatar
  • 11
2 votes
1 answer
86 views

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 ...
user371231's user avatar
  • 2,511
2 votes
0 answers
46 views

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 ...
Aericura's user avatar
  • 291

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