Questions tagged [p-groups]
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151 questions
18 votes
0 answers
570 views
Counting pairs of subgroups of $\mathbb{Z}^n$ whose intersection has given prime power index
Let $p$ be a prime, $n \geq 1$, and $F(x) := \prod_{i=0}^{n-1}(1-p^ix) = (1-x)(1-px)\cdots(1-p^{n-1}x)$. Claim: For each $k \geq 1$, the number of ordered pairs $(G,H)$ of subgroups of $\mathbb{Z}^n$ ...
- 25.9k
11 votes
2 answers
551 views
Extending scalars of p-groups
$\newcommand{\Z}{\mathbb Z}\newcommand{\F}{\mathbb F}$I would like to define a procedure which turns a finite $p$-group $G$ together with a field $K$ of characteristic $p$ (or even, in fact, any ring ...
- 535
1 vote
0 answers
73 views
A rank-selected generalization of a theorem of Kulakoff on $p$-groups
It is an elementary fact, due to P. Hall in 1933, that the number of subgroups of order $k$ of a $p$-group of order $p^n$, $0\leq k\leq n$, is congruent to $1$ modulo $p$. A. Kulakoff, Math. Ann. 104 (...
- 53.6k
3 votes
0 answers
157 views
Other known 5-groups of rank 2 and exponent 5?
I am interested in the known details of the other finite quotients of the free Burnside groups $B(m,n)$ of rank $m$ and exponent $n$. GAP's ANUPQ package has for example ...
- 131
2 votes
1 answer
182 views
Finite $p$-groups all of whose abelian subgroups are $2$-generated
It is well known that a finite $p$-group in which all abelian subgroups are cyclic (equivalently, in which $\Omega_1(G)$ has order $p$), is either cyclic or a generalized quaternion group. Does there ...
- 133
1 vote
1 answer
214 views
What is a torsion-complete abelian $p$-group?
I am trying to learn something about torsion-complete abelian $p$-groups. My reference is Fuchs, Infinite Abelian Groups, Vol II (Academic Press 1973). On page 15 he gives the setup. $B_n$ is a direct ...
- 561
12 votes
2 answers
427 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 ...
- 161k
2 votes
0 answers
77 views
Homomorphism to a finite p-group/Lie ring Q: estimate on |Q|
Let $L$ be a finitely generated, torsion-free nilpotent group. Furthermore, assume it is also a Lie ring, i.e. a Lie algebra but over $\mathbb{Z}$. The correspondance between $L$ as a group and $L$ as ...
- 61
3 votes
1 answer
203 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 ...
- 847
1 vote
0 answers
145 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 ...
- 133
8 votes
1 answer
264 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 ...
- 375
8 votes
1 answer
403 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 ...
- 375
4 votes
2 answers
302 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 $S$ ...
- 193
1 vote
0 answers
136 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 ...
- 287
3 votes
1 answer
193 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
235 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
91 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
180 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
104 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
134 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 |\...
- 387
1 vote
0 answers
130 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
239 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
193 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
106 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 ...
- 281
6 votes
2 answers
230 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 $\...
- 2,549
0 votes
0 answers
82 views
Existence of maximal topologically characteristic subgroup of infinite index of pro-$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 ...
- 847
-4 votes
1 answer
166 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
409 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
207 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 ...
- 1,051
7 votes
2 answers
426 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 ...
3 votes
0 answers
171 views
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 ...
- 967
5 votes
1 answer
253 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 ...
- 643
2 votes
0 answers
107 views
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 ...
- 391
1 vote
0 answers
206 views
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 ...
- 391
3 votes
1 answer
559 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
337 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 $< p$,...
- 3,770
5 votes
1 answer
337 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 ...
- 357
20 votes
1 answer
901 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 ...
- 4,236
2 votes
1 answer
318 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
226 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,...
- 4,791
1 vote
1 answer
316 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 ...
- 4,583
5 votes
0 answers
214 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 ...
- 66.6k
8 votes
1 answer
598 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
154 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
174 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^{}}}) ...
- 391
3 votes
0 answers
223 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 ...
- 391
1 vote
0 answers
162 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 ...
- 391
1 vote
0 answers
107 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-...
- 391
6 votes
1 answer
765 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 ....
- 391
0 votes
1 answer
169 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^{}}) ...
- 391