Questions tagged [p-groups]

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combinatorial way to count representatives of conjugacy class of elements of ord 5

I am trying to find a representative of each conjugacy class of order 5 elements in PGL$_6$($\mathbb C$). Let $r$ in $\mathbb C$ such that $r^5 = 1$ and [ ] denote modular the center of GL$_6(\mathbb ...
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3 votes
0 answers
69 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 ...
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  • 140
4 votes
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140 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 ...
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  • 213
2 votes
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54 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 ...
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1 vote
0 answers
110 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 ...
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3 votes
1 answer
208 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 ...
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  • 183
5 votes
1 answer
159 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 $< p$,...
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5 votes
1 answer
198 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 ...
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18 votes
1 answer
777 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 ...
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2 votes
1 answer
261 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 ...
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4 votes
0 answers
144 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,...
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1 vote
1 answer
205 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 ...
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5 votes
0 answers
151 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 ...
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8 votes
1 answer
500 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 ...
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1 vote
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....
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1 answer
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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^{}}}) ...
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2 votes
0 answers
131 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 ...
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1 vote
0 answers
110 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 ...
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1 vote
0 answers
59 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-...
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2 votes
0 answers
191 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 ....
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0 votes
1 answer
116 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^{}}) ...
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1 vote
1 answer
141 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^{}}}) ...
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1 vote
0 answers
55 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 $G$ ...
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3 votes
1 answer
220 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 $2n$ ...
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0 votes
1 answer
87 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 ...
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1 vote
0 answers
60 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 $...
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3 votes
0 answers
86 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 ...
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1 vote
0 answers
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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 ...
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1 vote
0 answers
57 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 $KG$, ...
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0 votes
0 answers
302 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(...
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6 votes
1 answer
117 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, ...
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1 vote
0 answers
153 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(...
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4 votes
1 answer
145 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 ...
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4 votes
0 answers
142 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 ...
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  • 4,219
3 votes
1 answer
179 views

For a pro-p, profinite group, abelianization being finitely generated is the same as being topologically finitely generated

I remember reading (without proof) that for $\Gamma$ a profinite, pro-$p$ group, the following are equivalent: 1) Every open subgroup $\Gamma_0$ is topologically finitely generated. 2) The ...
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  • 6,782
10 votes
0 answers
312 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 $\varphi(n)=n$? ...
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1 vote
0 answers
29 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 ...
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1 vote
0 answers
160 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 ...
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1 vote
1 answer
363 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 $...
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  • 457
2 votes
0 answers
73 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 ...
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4 votes
1 answer
167 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 ...
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2 votes
1 answer
170 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 $x\in Z_2(G)$? ...
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  • 99
1 vote
0 answers
130 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 ...
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  • 457
3 votes
1 answer
202 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/...
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  • 457
-2 votes
1 answer
147 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\...
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1 vote
0 answers
197 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 ...
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  • 457
1 vote
0 answers
28 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 $\Omega(P) \subset Z(P)$). ...
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0 votes
2 answers
850 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 ...
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  • 457
6 votes
1 answer
743 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 $q\neq p$. ...
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  • 117
2 votes
1 answer
728 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 ...
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