# Questions tagged [p-groups]

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140
questions

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### Maximal subgroups of finite abelian $2$-groups

Suppose $G$ is a finite abelian $2$-group, and $S$ is a subset of $G$, $\langle S\rangle=G$,$S^{-1}=S$,$e\notin S$. How to determine whether there exists a maximal subgroup $M$ of $G$, such that $S$ ...

0
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0
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92
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### Groups $P$ of order $p^5$ with $\Omega_1(P)=P$

I have been working with (particular) groups $P$ of order $p^5$. In fact, the ones that interest me the most are those that satisfy $$\langle x\in P\mid x^p=1\rangle=:\Omega_1(P)=P.$$ After a search ...

0
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0
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### Finite $p$-groups of maximal class whose generators have order $p$

Let $G$ be a finite $p$-group of maximal nilpotency class that is not cyclic of order $p^2$. Then $G$ is $2$-generated, say $G=\langle a,b\rangle$. Is there a classification in the case when $a^p=b^p=...

1
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0
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185
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### Prove that every group $G$ with $p^n$ ($n\ge4$) elements and center with $p$ elements has an abelian subgroup of order $p^3$ [closed]

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 ...

3
votes

0
answers

70
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### Admissibility of Ulm's invariants

Let $G$ be a reduced abelian $p$-group. We set $G_0=G$. Let $\alpha$ be an ordinal. Inductively, if $\alpha=\beta+1$ is a successor ordinal, we define
$$G_{\alpha}=pG_{\beta}.$$
If $\alpha$ is a limit ...

0
votes

0
answers

45
views

### Dimension inequality for primary groups

Let $p$ be a prime number and $G$ an abelian group.
The group $G$ is said to be $\textbf{primary}$ if every element of $G$ has order power of $p$.
For every natural number $n$, we define
$$\ker(p^n)=\{...

1
vote

1
answer

113
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### 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 ...

0
votes

0
answers

88
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### Invariants of primary groups

In Kaplansky's book "Infinite Abelian Groups", an abelian group $G$ is called primary if every element has order power of $p$ for some fixed prime number $p$. It is well-known that every ...

0
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0
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87
views

### The relation between two characteristic subgroups in finite p-group

Suppose $G$ is a finite $p$-group. Let
\begin{align*}
\mho_{1}(G)=\langle a^p\mid a\in G\rangle,\quad\Omega_{1}(G)=\langle a\in G\mid a^p=1\rangle.
\end{align*}
There are examples such that $|G|\leq |\...

1
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0
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115
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### Is this class of $p$-groups large?

Call a $p$-group $G$ good if for each subgroups $H, H_1, H_2\subseteq G$ for which $H_1\subseteq H$, $H_2\subseteq H$, $|H_1| = |H_2| = |H|/p$, $H_1\not= H_2$, $H'\not=\{e\}$ holds we have that there ...

6
votes

1
answer

204
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### 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?
$|H| = |G|/p$.
$c(H)\geq c(G) - 1$.

5
votes

1
answer

153
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### 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?

0
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0
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### Name of the power of the exponent of a $p$-group

Is there a name for the power of the exponent of a $p$-group? So, if $\mathrm{exp}(G):=\max\lbrace o(g)|g\in G\rbrace=p^k$ for some $k\in\mathbb{N}$, is there a name for the $k$? Additionally, is ...

6
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2
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185
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### 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 $\...

0
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0
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49
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### Existence of maximal topologically characteristic subgroup of infinite index of pro-$p$ groups

Let $G$ be a topologically finitely generated infinite pro-$p$ group. Suppose that $G$ is not just-infinite. Does the group $G$ always have a maximal topologically characteristic subgroup of infinite ...

-4
votes

1
answer

135
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### Exponential order of unipotent elements in an endomorphism ring of abelian groups

$\DeclareMathOperator\End{End}\newcommand{\Id}{\mathrm{Id}}$Let $E=\End(I)$ be the endomorphism ring of the abelian group $I$.
We have the following statement for $B\in E$, $p$ a prime number and $r$ ...

6
votes

1
answer

371
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### 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 ...

6
votes

1
answer

165
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### 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 ...

6
votes

2
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311
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### 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 ...

3
votes

0
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126
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### The intersection of the kernels of the real valued irreducible characters of a 2-group

For a $2$-group $P$ (that is, $|P|$ is a power of 2) let $K$ be the intersection of the kernels of the real-valued irreducible characters of $P.$ If the center $Z$ of $P$ is elementary abelian, then ...

4
votes

0
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159
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### The rank of indecomposable finite abelian 2-group

$\DeclareMathOperator\rank{rank}$Let $P$ be a finite $p$-group. The rank of $P$ is $\log_{p}|P/\Phi(P)|$ where $\Phi(P)$ is the Frattini subgroup of $P$, we write $\rank(P)=\log_{p}|P/\Phi(P)|$.
Let a ...

2
votes

0
answers

69
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### Wedderburn decomposition of wreath product of cyclic p-groups

Let $G$ be wreath product of cyclic group of prime order $p$ by itself, i.e. $G=C_p \wr C_p$, where the action of $C_p$ is taken as cyclic permutation on generators of first $p$ cyclic groups. Can we ...

1
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0
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147
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### Wedderburn decomposition of semisimple group algebras

Let $G$ be a finite $p$-group. What can we say about the Wedderburn decomposition of the group algebra $FG$? Here $F$ is a finite field of characteristic co-prime to $p$. Can we say something in the ...

3
votes

1
answer

433
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### Structures of subgroups of a finite abelian p-group

$\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $G=\mathbb{Z}/p^{i_1}\times\cdots\times\mathbb{Z}/p^{i_r}$ with $i_1\leq\ldots\leq i_r$ be a finite abelian $p$-group. Then there can be many ...

5
votes

1
answer

222
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### 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 $< p$,...

5
votes

1
answer

252
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### 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 ...

19
votes

1
answer

834
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### 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 ...

2
votes

1
answer

288
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### Status of a conjecture of Thompson

Let $ S $ be a finite group. Denote by $\mathcal{B}_0(S)$ the set of the subgroups $H$ of $S$ satisfying $|H:H'| > |K:K'|$ for every proper subgroup $K$ of $H$ ($H'$ denotes the drived subgroup of ...

4
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0
answers

177
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### On 2-groups of exponent 4 and class 2

Suppose A is a 2-group with the following properties:
$\lvert A \rvert = t^3$ with $t$ some even power of $2$;
$A$ and $Z(A)$ (the center of $A$) are of exponent $4$;
$\lvert Z(A) \rvert = t$ and $[A,...

1
vote

1
answer

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### How many elements of each order are there in this $p$-group? [closed]

Let $G$ be a countable Abelian $p$-group which equals a direct sum of at most countably many finite cyclic groups and at most countably many copies of the Prüfer $p$-group, where these finite cyclic ...

5
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0
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### 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 ...

8
votes

1
answer

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### 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 ...

1
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1
answer

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### Infinite pro-$p$ group of finite solvable length and finite coclass

I was reading about infinite pro-$p$ groups of finite coclass from the book "The Structure of Groups of Prime Power Order" by Leedham-Green and McKay. I asked this question in math....

3
votes

1
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133
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### Permutation representation of a finite $p$-group

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...

3
votes

0
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147
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### Units in group rings

Let $F$ be any field with $p$ elements and $G$ be any finite $p$-group, combining together they form a group ring $FG$. And $V(FG)$ denotes group of units of coefficient-sum equal to 1 in $FG$. We ...

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126
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### Structure/description of a finitely presented group

I am unable to see the structure of the following finitely presented group.
$$\langle a,b,c,d : [a,b]=c=a^p,\ [c,b]=c^p=d^p,\ b^{p^2}=c^{p^2}=1 \rangle$$
I have tried in GAP, but it is not showing any ...

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0
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86
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### On isoclinism classes of finite p-groups

With reference to
James, Rodney, The groups of order (p^6) ((p) an odd prime)., Math. Comput. 34, 613-637 (1980). ZBL0428.20013., My question is can we get isoclinism class $\phi_2$ for a finite p-...

6
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1
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557
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### 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 ....

0
votes

1
answer

135
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### Faithful representation of group of order $p^4$

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book, "Theory of groups of finite order". The group ($\mathbb{Z}_{p^{2}}\rtimes \mathbb{Z}_{p^{}}) ...

1
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1
answer

183
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### Presentations of groups of order $p^4$

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...

1
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0
answers

59
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### Finite $p$-groups of co-class $3$, class at least $4$ and some controlled generator growth

I am trying to prove the following comment (Ref. https://link.springer.com/article/10.1007/s00605-016-0938-5 Page-684, Rmk3.2):
Let $G$ be a finite $p$-group of co-class $3$, class $\geq 4$. Then $G$ ...

3
votes

1
answer

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### Direct proof (or reference) that a given $p$-group is extra-special

Writing a paper on algebraic surfaces, I was led to consider the finite group $\mathsf{H}(A)$ whose presentation is the following.
I start with an anti-symmetric matrix $A=(a_{ij})$ of order $2n$ ...

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votes

1
answer

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### An outer automorphism of a 3-group of maximal class

Let $G$ be a finite 3-group of maximal class. The center $Z(G)$ contains two elements other than the identity. Does there exist an endomorphism of $G$ that maps one of them to the other?
This is true ...

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0
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64
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### When is the following preorder on the set of central elements of order 2 a total preorder?

Let $G$ be a finite 2-group. Denote by $S$ the set of central elements of $G$ of order exactly $2$. The relation $a\leq b$ iff there is an endomorphism of $G$ sending $b$ to $a$ defines a preorder on $...

3
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0
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93
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### A question to the derived length in modular group algebras

Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group ...

1
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0
answers

23
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### Exponents in unit groups of modular group algebras

Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group ...

1
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0
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### Number of conjugacy classes of unit groups of modular group algebras

Let $n$ be a natural number, $p$ a prime number, $G$ a finite $p$-group and $K$ a finite field with $p^n$ elements. We focus on the group $1+J(KG)$, where $J(KG)$ is the Jacobson radical of $KG$, ...

0
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334
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### Normal subgroups of $p$-groups

I was reading Professor Yukov Berkovich' paper "On Subgroups of Finite $p$-groups" when I stumled upon the following theorem:
Let $G$ be a nonabelian $p$-group with cyclic Frattini subgroup, $|\Phi(...

6
votes

1
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141
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### $p$-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, ...

1
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0
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176
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### Computing the class-preserving automorphism group of finite $p$-groups

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