All Questions
Tagged with gr.group-theory p-groups
121 questions
8
votes
0
answers
200
views
Logarithm of a $p$-group in $\mathrm{GL}_n(p)$
$\def\GL{\operatorname{GL}}\def\ZZ{\mathbb{Z}}\def\FF{\mathbb{F}}\def\Id{\mathrm{Id}}\def\fu{\mathfrak{u}}$Let $p$ be prime, let $n<p$, let $U_n(\FF_p)$ be the group of $n \times n$ upper ...
- 156k
3
votes
1
answer
133
views
Any Sylow pro-$p$ subgroup of a topologically finitely generated profinite group is also topologically finitely generated?
It's proved in Oltikar and Ribes - On prosupersolvable groups that any Sylow pro-$p$ subgroup of a topologically finitely generated prosupersolvable group is also topologically finitely generated. It ...
- 667
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
217
views
Bilinear forms on abelian $p$-groups: Images of weak metabolizers in the Frattini quotient
Suppose $G$ is a finite abelian $p$-group for $p$ an odd prime and $b : G \times G \to \mathbf{Q}/\mathbf{Z}$ is a non-degenerate symmetric bilinear form. A subgroup $H \leq G$ is a weak metabolizer ...
- 365
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
0
answers
198
views
Prove that every group $G$ with $p^n$ ($n\ge4$) elements and center with $p$ elements has an abelian subgroup of order $p^3$ [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
views
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 ...
- 31
1
vote
1
answer
132
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 ...
0
votes
0
answers
90
views
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 ...
- 31
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
1
vote
0
answers
116
views
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 ...
- 291
6
votes
1
answer
213
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?
$|H| = |G|/p$.
$c(H)\geq c(G) - 1$.
- 291
5
votes
1
answer
159
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?
- 291
0
votes
0
answers
82
views
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 ...
- 91
0
votes
0
answers
54
views
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 ...
- 667
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
5
votes
1
answer
277
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 ...
- 347
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
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
1
vote
1
answer
276
views
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 ...
- 4,597
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
0
answers
180
views
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 ...
- 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
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
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
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
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
6
votes
1
answer
154
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, ...
- 26.5k
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
4
votes
1
answer
152
views
Do the class vector and character vector of a $p$-group determine each other?
To a finite $p$-group, we can associate two vectors $(v_0,v_1,\dotsc)$:
The class vector - $v_i$ is the number of conjugacy classes of order $p^i$.
The character vector - $v_i$ is the number of ...
- 5,654
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
1
vote
0
answers
31
views
Defect of subnormality in unit groups of modular group algebras
Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $char(K)=p$. It is well-known that the group $1+rad(KG)$ is a p-group containing $G$. $G$ is normal in $1+rad(KG)$ if and ...
- 581
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
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 ...
2
votes
1
answer
177
views
a question about finite 2-group
Please help me about the following question:
Suppose $G$ is a finite 2-group and $x\in Z(M)\setminus Z(G)$ for some maximal subgroup $M$ of $G$ such that $x^2\in Z(G)$, is it true $x\in Z_2(G)$?
...
- 99
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