All Questions
Tagged with finite-groups p-groups
82 questions
1
vote
0
answers
109
views
Center of factors of a finite $p$-group, obtained from a minimal normal subgroup
throughout a research problem about finite $p$-groups,
I have a challenge as follows,
Let $G$ be a finite non-abelian $p$-group, where $p$ is odd and $Z(G)$ is non-cyclic.
($Z(G)$ denotes the center ...
- 111
8
votes
1
answer
353
views
Structure of a single automorphism of a finite abelian p-group
A finite abelian $p$-group $H$ is homogenous when it is the direct sum of cyclic groups of the same order $p^r$, i.e. $H \cong \big(\mathbb{Z}/p^{r}\mathbb{Z}\big)^{e}$. Every finite abelian $p$-group ...
- 365
4
votes
2
answers
227
views
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 $S$ ...
- 173
1
vote
0
answers
105
views
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 ...
- 245
3
votes
1
answer
149
views
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=...
- 33
1
vote
1
answer
132
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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
96
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 |\...
- 171
-4
votes
1
answer
136
views
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 $r$ ...
- 93
6
votes
1
answer
377
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 ...
- 933
6
votes
1
answer
172
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 ...
- 1,021
6
votes
2
answers
351
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 ...
5
votes
1
answer
211
views
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 ...
- 613
3
votes
1
answer
474
views
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 ...
- 321
5
votes
1
answer
247
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 $< p$,...
- 3,230
19
votes
1
answer
847
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 ...
- 3,932
2
votes
1
answer
295
views
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 ...
- 31
4
votes
0
answers
186
views
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,...
- 4,547
8
votes
1
answer
536
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 ...
- 405
1
vote
1
answer
102
views
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....
- 243
3
votes
1
answer
143
views
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^{}}}) ...
- 381
1
vote
0
answers
128
views
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 ...
- 381
1
vote
0
answers
88
views
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-...
- 381
6
votes
1
answer
621
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 ....
- 381
0
votes
1
answer
143
views
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^{}}) ...
- 381
1
vote
1
answer
196
views
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^{}}}) ...
- 381
1
vote
0
answers
60
views
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 $G$ ...
- 405
3
votes
1
answer
269
views
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 $2n$ ...
- 66.3k
0
votes
1
answer
101
views
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 ...
user145520
1
vote
0
answers
65
views
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 $...
user145520
3
votes
0
answers
95
views
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 ...
- 581
1
vote
0
answers
23
views
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 ...
- 581
1
vote
0
answers
67
views
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 $KG$, ...
- 581
0
votes
0
answers
345
views
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(...
- 415
1
vote
0
answers
193
views
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(...
- 41
7
votes
0
answers
405
views
How can I get my hands on McKay's "Finite p-groups" lecture notes?
How can we find Susan 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 ...
- 4,425
11
votes
1
answer
502
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 $\varphi(n)=n$?
...
- 11.1k
1
vote
0
answers
168
views
about a strange property of p-groups of maximal class
I am trying to look for a finite $p$-group of maximal class of order at least $p^{2p+1}$ exponent at least $p^3$ which possibly has the following property :
If s is an element in $G-G_1$ ($G_1$ is ...
- 405
2
votes
1
answer
483
views
Certain $p$-group with cyclic center
Let $G$ be a finite $p$-group of derived length $d$, which is not a Dedekind group.
(i.e., possesses at least one non-normal subgroup).
Let $G^{(d-1)}$ be the unique normal subgroup of $G$ of order $...
- 487
2
votes
0
answers
98
views
p-group of maximal class
I am trying to prove that if $G$ is a $p$-group of maximal class and order $p^4$ ($p$ odd), then its unique two-step centraliser $G_1=C_G([G,G])$ is of the form $C_{p^2}\times C_p$. It is clear from ...
- 405
4
votes
1
answer
207
views
Central extensions of Suzuki 2-groups
Recall the definition of the finite Suzuki 2-groups: These are finite non-abelian 2-groups that contain more than one involution such that a solvable group of automorphisms permutes the involutions ...
1
vote
0
answers
145
views
Conjugacy classes of non-normal subgroups of a finite $p$-group
Let $G$ be a finite $p$-group of derived length $d$ and nilpotency class $c$. Suppose that $G$ is not a Dedekind group (i.e., possesses at least one non-normal subgroup). Suppose that $G^{(d-1)}$ has ...
- 487
4
votes
1
answer
332
views
The nilpotency class and the derived length of a $p$-group
Let $G$ be a finite $p$-group of nilpotency class $c$ and of derived length $d$.
As is well known, we have $d\leq \lfloor\log_2 c\rfloor+1$,
(https://groupprops.subwiki.org/wiki/...
- 487
-2
votes
1
answer
162
views
Number of subgroups of a group of orders $p^3$ [closed]
Let $p$ be a prime number. Is there a formula for the number of subgroups of
$$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p^2\mathbb{Z}$$
$$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}\times\...
- 39
1
vote
0
answers
239
views
Cyclic subgroups of finite $p$-groups
Let $G$ be a finite non-Dedekind $p$-group with non-cyclic center, where $p$ is an odd prime.
By $[\langle x\rangle]_G=\{g^{-1}\langle x\rangle g\ |\ g\in G\},$
I mean the conjugacy class of the ...
- 487
1
vote
0
answers
39
views
Bounding the exponent of finite $p$-groups with normalizer conditions on cyclic subgroups
Suppose $P$ is a non-cyclic finite $p$-group satisfying the following two conditions:
All cyclic subgroups of order $p$ in $P$ are normal (this is equivalent to saying that $\Omega(P) \subset Z(P)$).
...
0
votes
2
answers
956
views
Existence of a cyclic non-normal subgroup in a $p$-group
Let $G$ be a finite non-abelian $p$-group, where $p$ is an odd prime,
$N$ be a normal subgroup of $G$ of order $p$, where $\frac{G}{N}$ is non-abelian.
Does there exist an element $g\in G$ such that ...
- 487
2
votes
1
answer
860
views
On $ p $-groups with at least one element of order $ p^{2} $
Let $G$ be a finite non-abelian $p$-group such that $G$ contains at least an element of order $p^2$ and for every nontrivial normal subgroup $N$, $G/N$ has not any elements of order p^2 and G/Z(G) is ...
- 21
0
votes
1
answer
198
views
p-groups with special property on its centralizers
Thanks for any help or comment.
Suppose $G$ is a finite non-abelian p-group. Suppose $G$ has a proper non-abelian subgroup $M$ such that for every non-central element $x\in M$, $C_G(x)\subseteq M$. ...
- 71
32
votes
3
answers
3k
views
Is there a nice explanation for this curious fact about cyclic subgroups?
Here's something that I noticed that quite surprised me.
Let $G$ be a finite abelian group. Consider the following expression.
$$
\nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H|
$$
It ...
- 6,290
3
votes
2
answers
268
views
p-groups as finite union of disjoint normal abelian subgroups
I was interested in knowing if groups with following property have been studied( like what can be said about structure of the group) :
"$G$ can be written as disjoint union of a given number of ...
- 31