Questions tagged [p-groups]

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Richness of the subgroup structure of p-groups

Given a prime $p$ and $n \in \mathbb{N}$, let $f_p(n)$ be the smallest number such that there is a group of order $p^{f_p(n)}$ which all groups of order $p^n$ embed into. What is the asymptotic growth ...
Stefan Kohl's user avatar
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32 votes
3 answers
3k views

Is there a nice explanation for this curious fact about cyclic subgroups?

Here's something that I noticed that quite surprised me. Let $G$ be a finite abelian group. Consider the following expression. $$ \nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H| $$ It ...
Simon Rose's user avatar
  • 6,240
21 votes
0 answers
566 views

p-groups as rational points of unipotent groups

Is it true that every finite p-group can be realized as the group of rational points over $\mathbb{F_p}$ of some connected unipotent algebraic group defined over $\mathbb{F_p}$? For abelian p-groups, ...
Georges's user avatar
  • 221
19 votes
1 answer
829 views

Groups with a unique lonely element

Does there exist a finite group $G$ of order greater than two containing a unique element $g$ such that $$ g\notin\langle x\rangle \hbox{ for all $x\in G\setminus\{g\}$ ?} $$ Or we have another ...
Anton Klyachko's user avatar
13 votes
1 answer
440 views

Variety of nilpotent Lie algebras or $p$-groups

Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either: 1) Let $\mathcal{L}$ ...
YCor's user avatar
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11 votes
1 answer
484 views

Is the norm element characteristic in modular group rings?

Let $G$ be a finite $p$- group and let $\varphi$ be an automorphism of $\mathbb{F}_pG$ as $\mathbb{F}_p$-algebras and let $n = \sum_{g\in G} g$ be the norm element. Does it follow that $\varphi(n)=n$? ...
HenrikRüping's user avatar
10 votes
3 answers
942 views

faithful unipotent representations of (finite) $p$-groups

The title pretty much summarizes the question: does every $p$-group have a faithful unipotent representation (with coefficients in $\mathbb{F}_p$ or some finite extension thereof)?
Igor Rivin's user avatar
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10 votes
5 answers
953 views

Automorphism Group of a p-group : Looking for a Reference

In the following post by DavidLHarden : See Here He quoted the following claim: "There is a theorem that says that if $p$ is a prime and $|G|=p^n $ , then $|AutG| $ divides $ \Pi_{k=0}^{n-1} (p^{n}-...
Jason Mraz's user avatar
10 votes
3 answers
6k views

Number of Normal subgroups In a p-Group

Dear all, Does someone know of any paper/method that enables us counting/estimating the number of normal subgroups of some p-group of order $p ^n $ ($ n$ is some natural number ? ) . Is there anyway ...
Jason Mraz's user avatar
10 votes
4 answers
1k views

Classification of automorphism groups of groups of order $p^4$

For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of p-...
Giuliano Bianco's user avatar
10 votes
1 answer
569 views

Maximal subgroups of a certain finite 2-group

The following came up in a problem on reconstruction of digraphs. I determined enough about the answer to satisfy the application completely, but still I am curious to know what the complete solution ...
Brendan McKay's user avatar
9 votes
2 answers
374 views

Which finite p-groups occur as commutators of finite p-groups?

Let $p$ be a prime number. For which finite $p$-groups $H$ is there a finite $p$-group $G$ such that $[G,G] \cong H$?
Pablo's user avatar
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9 votes
1 answer
281 views

p-groups such that the center is contained in many cyclic subgroups

I'm looking for examples of $p$-groups $G$ with the following three properties: the center of $G$ is $\mathbb{Z}/p$, and $G^{\text{ab}} = (\mathbb{Z}/p)^n$ for some $n$, and for every $g \in G$ whose ...
Ligia's user avatar
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8 votes
8 answers
4k views

classification of $p$-groups

I have two questions regarding to $p$-groups. A $p$-group $G$ is said to be extraspecial of $G'=Z(G)$ has order $p$. Hence extraspecial groups are examples of $p$-groups with cyclic center. Of ...
M.B's user avatar
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8 votes
2 answers
459 views

Uniform-in-p classification* of p-groups of order p^n for each fixed n?

To what extent is there/can there be a description that is uniform in p (for p sufficiently large) of the p-groups of order $p^n$, for each fixed n? Note 1: I used the word "description" rather than ...
Joshua Grochow's user avatar
8 votes
2 answers
1k views

Representation theory of a finite p-group over a field of characteristic p: dim of invariants =1 => dim of coinvariants = 1?

I am trying to understand the proof of Proposition 4 in S. Ullom, Integral normal bases in Galois extensions of local fields, Nagoya Math. J. Volume 39 (1970), 141-148. The PDF is available here: http:...
Henri Johnston's user avatar
8 votes
2 answers
294 views

Is the free abstract group residually of rank d > 2?

Let $d \geq 2$ be an integer, and let $\mathcal{F}_d$ be the family of finite groups such that $G \in \mathcal{F}_d$ if and only if every subgroup of $G$ can be generated by at most $d$ elements. Is ...
Pablo's user avatar
  • 11.2k
8 votes
1 answer
532 views

Constructing a group of order $2187=3^7$

I am trying to look for the $2$-generated groups of order $3^7$ and class $4$ all whose upper central series quotients are elementary abelian of order 9 except the center which has order $3$. A small ...
Siddhartha's user avatar
7 votes
3 answers
893 views

Characters of p-groups

Berkovich mentioned the following result of Mann in his book on p-groups: The number of nonlinear irreducible characters of given degree in a p-group is divided by p-1. Do you know any reference for ...
Amin's user avatar
  • 307
7 votes
3 answers
616 views

p-group with large center

Is there any characterization for $p$-groups of order greater than $p^3$ which center has index $p^2$? (One group whit this property if $M(p^n)$)
Hamid Shahverdi's user avatar
7 votes
1 answer
366 views

Is $[729,57]$ a Sylow $3$-subgroup of some well-known group?

Let $G$ be the group $[729,57]$, using GAP's notation. I have so far two descriptions of the group: a presentation an embedding (not surjective!) of the group into a Sylow $3$-subgroup of the unit ...
user avatar
6 votes
4 answers
3k views

Center of p-groups

Is it true that any abelian $p$-group (not necessarily finite) is the center of a $p$-group and of index $p^2$?
i. m. soloveichik's user avatar
6 votes
3 answers
908 views

Union of conjugates of a subgroup

Let $G$ be a finite group, $H \leq G$ a proper subgroup. It is well known that the union of the conjugates of $H$ does not cover $G$. I would like to know of more precise results (even in special ...
Pablo's user avatar
  • 11.2k
6 votes
1 answer
163 views

Is the largest normal abelian subgroup of a finite 2-group $P$ of order at least the square root of the order of $P$?

Let $G$ be a group of order $2^n$. Does $G$ have a normal abelian subgroup of order at least $2^{n/2}$? (This is true, via computations in GAP, for $n \le 8$. The question is similar to one posed ...
Ken W. Smith's user avatar
6 votes
1 answer
204 views

Is there a subgroup of a non-abelian $p$-group $G$ with a large nilpotency class?

Let $G$ be a non-abelian $p$-group ($p\ne2$). Does there exist a group $H\subset G$ such that both 1, 2 are satisfied? $|H| = |G|/p$. $c(H)\geq c(G) - 1$.
solver6's user avatar
  • 291
6 votes
2 answers
305 views

Differences between $p$-groups and $q$-groups

First, let me include the same disclaimer that goes in the first line of any article I write: all groups considered herein are finite. Academically, I work with connecting the arithmetic structure of ...
6 votes
1 answer
295 views

Maximal cyclic quotient of a $p$-group

Let $G$ be a finite abelian $p$-group, $p$ a prime. I say that a pair $(G',\varphi)$ is a maximal cyclic quotient (please excuse me if this definition already exists and refers to a different concept) ...
Ferra's user avatar
  • 509
6 votes
1 answer
368 views

Finite 2-groups with $(ab)^{2}=(ba)^{2}$

There exist nonabelian finite 2-groups $G$ with the property $(A2)$ : for every $a,b\in G$, $(ab)^{2}=(ba)^{2}$. An example of a such group is given by the quaternion group $Q_{8}$ of order 8. Is ...
Rajkarov's user avatar
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6 votes
1 answer
533 views

On classifying groups of order $p^5$

Can someone suggest me some source where the author has classified all non-isomorphic groups of order $p^5$ ? I need complete classification (not upto isoclinism), and also in finitely presented form ....
HIMANSHU's user avatar
  • 381
6 votes
1 answer
137 views

$p$-groups with isomorphic subgroup lattices

Given two non-abelian finite p-groups $P_1$ and $P_2$ of the same order that are not isomorphic. Can $P_1$ and $P_2$ have isomorphic subgroup lattices? (I'm not experienced with group theory, ...
Mare's user avatar
  • 25.8k
6 votes
2 answers
184 views

Agemo-of-agemo inclusions for p-groups

For a finite $p$-group $G$, let $\mho_i(G)$ denote the subgroup generated by $p^i$-powers of elements of $G$. It is well-known that $\mho_i(\mho_j(G))$ can differ from $\mho_j(\mho_i(G))$ and from $\...
grok's user avatar
  • 2,489
6 votes
1 answer
825 views

Extra special p-groups

Let $P$ be an infinite extra special $p$-group for some prime $p$, namely, $Z(P)=P'=\Phi(P)$ and $P/Z(P)$ is infinite elementary abelian. Let $C$ be a Prufer $q$-group for some prime $q\neq p$. ...
W4cc0's user avatar
  • 137
6 votes
1 answer
675 views

Torsion in profinite groups

Is there a finitely generated profinite group $G$ with a closed subgroup of infinite index $K \leq G$ such that for every $g \in G$ there exists some $n \in \mathbb{N}$ for which $g^n \in K$ ? Can $G$...
Pablo's user avatar
  • 11.2k
6 votes
0 answers
103 views

Random pro-p groups via iterated uniformly random central extensions

Inspired by this question on math.se, I want to understand the following construction of a random pro-$p$ group: We want to construct an inverse system $$\cdots \xrightarrow{\alpha_i} G_i \...
user68822's user avatar
  • 381
6 votes
0 answers
192 views

Is there a Noetherian profinite group of infinite rank?

Is there a profinite group $G$ such that any closed subgroup $H \leq G$ is finitely generated, but there is no $n \in \mathbb{N}$ such that every closed subgroup of $G$ can be generated by at most $n$ ...
Pablo's user avatar
  • 11.2k
6 votes
0 answers
412 views

An example of a simple infinite 2-group

I've asked this question before on Mathematics, and they suggested me to ask here (Link). Is there an example of a simple infinite $2$-group? Informations If a $2$-group is Artinian I know that it ...
W4cc0's user avatar
  • 599
5 votes
3 answers
549 views

Normal abelian subgroups in p-groups

Given a group $G$, we denote by $T(G)$ the subgroup generated by all (maximal) normal abelian subgroups of $G$. Let define the series $(T_i(G))$ by $T_0(G)=1$ and $T_{i+1}(G)/T_i(G)=T(G/T_i(G)$, and $...
Yassine Guerboussa's user avatar
5 votes
1 answer
152 views

Do these $p$-groups have the same nilpotency class?

Let $G$ be a $p$-group, $\{e\}\not= H\subseteq G$ be a subgroup of $G$ such that $G' = H'$. Is it true that $c(G) = c(H)$, where $c(\cdot)$ denotes the nilpotency class of a group?
solver6's user avatar
  • 291
5 votes
1 answer
245 views

Number of subgroups of a $p$-group of index $p^k$

Let $p$ be a prime, let $n$ and $k$ be positive integers and let $G$ be a group of order $p^n$. Further, let $a_{p^k}$ denote the number of subgroups of $G$ of index $p^k$. If $a_{p^k}$ is greater ...
Leyli Jafari's user avatar
5 votes
1 answer
218 views

Finite solvable groups are generated by a nilpotent subgroup + K elements?

Is there a constant $K \in \mathbb{N}$ such that for every finite solvable group $G$, there exists a nilpotent subgroup $N \leq G$, and a subset $S \subseteq G$ with $|S| \leq K$, and $\langle N,S\...
Pablo's user avatar
  • 11.2k
5 votes
3 answers
374 views

Hall algebra for non-abelian $p$-groups?

According to WP article on Hall algebras one counts the number of abelian subgroups in an abelian group with fixed type of subgroup, group, quotient. Two things are claimed: These numbers are ...
Alexander Chervov's user avatar
5 votes
2 answers
747 views

Center of finite metabelian p-groups

$\DeclareMathOperator\rk{rk}$ Let $G$ be a finite metabelian $p$-group, i.e. the commutator subgroup $G'$ of $G$ is abelian. Then I ask myself under which conditions does the following hold: $$\tag{$*...
Tobias Bembom's user avatar
5 votes
1 answer
208 views

Local vs global nilpotence class (Lazard correspondence)

The Lazard Correspondence is often phrased (for simplicity) for $p$-groups of nilpotence class $c < p$, but it works more generally whenever every 3-generated subgroup has nilpotence class $< p$,...
Joshua Grochow's user avatar
5 votes
0 answers
190 views

Can an infinite abelian $p$-group be tall and thin?

Does there exist an abelian $p$-group $A$ with countable Ulm invariants and uncountable height? Here by height, I mean the minimal ordinal $\rho$ such that $p^\rho A$ is divisible [1]. For an ordinal ...
Tim Campion's user avatar
  • 60.6k
5 votes
0 answers
170 views

How can I get my hands on McKay's "Finite p-Groups" lecture notes?

The notes I'm talking about are these. I emailed Peter Cameron, but he has since moved to a different university, and has no copies himself. I also emailed the school manager at Queen Mary, but they ...
Steve D's user avatar
  • 4,335
5 votes
0 answers
293 views

A class 3 group of order 243

Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a (...
user avatar
4 votes
3 answers
2k views

Representation theory of p-groups in particular upper tringular matrices over F_p

Finite p-groups - have p^n elements by definition. According to WP there is rich structure theory. Question: How far is representation theory of p-groups is understood? In case this question is too ...
Alexander Chervov's user avatar
4 votes
3 answers
485 views

Molien for modular representations?

Let $G$ be a finite group, and let $k$ be a field whose characteristic divides $\left|G\right|$. Let $\rho:G\to \mathrm{End} V$ be a (finite-dimensional) representation of $G$ over $k$. Prove or ...
darij grinberg's user avatar
4 votes
1 answer
195 views

Finite p-groups and their fibered products

Is every finite $p$-group an epimorphic image of a fibered product of two finite $p$-groups which can be generated by $2$ elements?
Pablo's user avatar
  • 11.2k
4 votes
1 answer
419 views

Generators of p-groups

Let $G$ be a finite $p$-group. Since we can embed $Z_2(G)/Z(G)$ in $Hom(G,Z(G))$, we have $d_2 \leq d(G)d(Z(G))$; where $d_2(G)=d(Z_2(G)/Z(G))$ and $d(G)$ denotes the minimal number of generators of $...
Yassine Guerboussa's user avatar