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Questions tagged [p-groups]

Use with the (group-theory) tag. Refers to questions concerning finite groups of prime power order or infinite p-groups such as Prüfer groups, pro-p-groups, and Tarski monsters. This tag is not for p-adic number systems.

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1answer
129 views

Let $G$ be a non-abelian finite $p$-group, do we have $Z(G) \leq \Phi(G)$ in general?

Let $G$ be a non-abelian finite $p$-group. I wonder if $Z(G) \leq \Phi(G)$, where $\Phi(G)$ is the Frattini subgroup of $G$? I'd known that it is true for non-abelian groups of order $p^3$ but I ...
4
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2answers
79 views

Finding the number of subgroups of $\mathbb{Z}_{p^3} \oplus \mathbb{Z}_{p^2} $.

What is the number of subgroups of $\mathbb{Z}_{p^3} \oplus \mathbb{Z}_{p^2} $? I'm not really sure how to figure it out. I tried seeing subgroups of each $\Bbb Z_{p^n}$ but I'm not sure I'm going to ...
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0answers
41 views

If $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$

In 'Finite Groups' by Gorenstein, it is stated that if $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$′. The proof is the ...
1
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0answers
42 views

Intersection of all maximal subgroups of a finite group

Let $G$ be a group. Define $$F=\left\lbrace x\in G\;:\; \text{if } \left\langle S,x\right\rangle=G\text{ then }\left\langle S\right\rangle=G\right\rbrace.$$ I have that $F$ is a normal subgroup of $G$....
3
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1answer
74 views

Finding some explicit formula for $(ab)^n$ in any $a,b$ in a finite $p$-group.

If $G$ is a finite $p-$group, let $a$ and $b$ any two elements from $G$. Is there any formula for $(ab)^n$ involving $a^nb^n$ for any natural number $n$? That is, some formula like $(ab)^n = a^nb^...
1
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1answer
36 views

Normal subgroup $N$ of a p-group $G$ intersects $Z(G)$ nontrivially; What is wrong with the following trivial argument?

I'm trying to show that a normal subgroup $N$ of a p-group $G$ intersects $Z(G)$ nontrivially (please don't tell how to show it), but it seem it is quite a trivial question considering the following ...
2
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0answers
29 views

On $n$th class-preserving automorphism of finite $p$-group

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called an $n$th class-preserving if for each $x\in G$, there exists an element $g_x\in \gamma_n(G)$ ...
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0answers
18 views

Quick question about proving that either H is contained in K or K is contained in H.

Let G be a cyclic p-group with subgroups H and K. Prove that either H is contained in K or K is contained in H. I am looking at Alan Wang's answer, and I am a little confused. Why is it $H\leq K$ and ...
2
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1answer
40 views

About quotients of Lower Exponent$-p$ central Series

Let $G$ be a finite $p-$group of number of generators $d$ and exponent$-p$ class $c$, that is $c$ is the smallest integer satisfying $P_c(G) =1$ in the series $$ G=P_0(G) \geq ...\geq P_{i-1}(G)\geq ...
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0answers
24 views

Why is the $p-$multiplicator of a $p-$group is an elementary abelian $p -$group?

Let $G$ be a $p-$group that have the finite presentation $F/R$($F$ is a free group of rank $d$); The $p-$multiplicator of $G$ is defined by $$ G^* = R/[F,R]R^p $$ Why $G^*$ in an elementary abelian $p ...
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2answers
33 views

Error in proving subgroup is normal

Let $G$ be a $p$-group and let $H$ be a subgroup of $G$ of index $p$ ($p$ is prime). Prove that $H$ is normal. I have tried to prove it, but accidentally have proven the opposite by some error: Let ...
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0answers
33 views

Detailed usage of abelianisation and row-echelonisation?

Let $G=⟨x,y\mid[[x,y],x]=x^2,(xyx)^4,x^4,y^4,(yx)^3y=x⟩$ with $p=2$. I hope to show me in details how to compute $G/P_1(G)$(A polycyclic presentation for $G/P_1(G)$), where $P_1(G)$ is the second ...
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2answers
38 views

Example of non-abelian groups with these properties

I am looking for examples of non-abelian groups of arbitrarily large size with the following properties Have order $p^a$, where $a$ is a positive integer and $p$ is prime. Contain an abelian ...
3
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1answer
53 views

Subgroups of $G^n$, where $G$ is a $p$-group

Let $G$ be a finite $p$-group and let $n > 0$. Let $G^n$ be the direct product of $n$ copies of $G$. Are all subgroups of $G^n$ isomorphic to $H_1 \times \dotsm \times H_n$ for some subgroups $...
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0answers
31 views

Relationship between Frattini subgroup and the first Agemo subgroup of a prime-power order group

Let $G$ be a $p$-group for an odd prime $p$. The Frattini subgorup $\Phi(G)$ is defined as the intersection of all maximal subgroups or, equivalently, as the subgroup of non-generating elements, i.e. ...
2
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0answers
31 views

Bound on order of commutator subgroup of a $p$-group

I was reading an article where it is claimed that If $G$ is a finite $p$-group with $|G|=p^n$ and nilpotency class of $n-2$ where $n\ge 7$ then $p\le|Z(G)|\le p^2$ and $p^{n-3}\le |G'|\le p^{n-2}$....
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1answer
16 views

Any $p$-order subgroup is normal in a $pk$ group

I'm looking to prove that if $G$ is a group of order $pk$ where $p$ is prime and $p>k$, that any subgroup $K\leq G$ of order $p$ is normal in G. Does anybody have any hints or tips for proving ...
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2answers
88 views

Number of normal subgroups of order $p^s$ of a $p$-group

Let $G$ be a $p$-group. Show that the number of normal subgroups of $G$ that have order $p^s$ is $1$ $mod(p)$. I think I have to use Sylow theorems and the fact that every subgroup of order $p^s$ is ...
1
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1answer
27 views

Are these two semidirect products isomorphic?

Let $p$ be a prime. Then there is a nonabelian semidirect product of $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p$. There is also a nonabelian semidirect product of $\mathbb{Z}_p\oplus\mathbb{Z}_p$ and $\...
2
votes
1answer
64 views

$p$-group of odd order with all proper subgroups cyclic

If $p$ is an odd prime and $G$ is a group of order $p^r$, with $r\ge 3$, such that all proper subgroups of $G$ are cyclic, then is $G$ cyclic ?
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0answers
50 views

Automorphism group of finite $p$-group when automorphism group of its quotient is $p$-group

I was reading articles and books on automorphism group. It is always an interesting question to decide when the automorphism group is a $p$-group. In this regard my question is Let $\gamma(H)$ be ...
1
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0answers
32 views

Upper central series, semidirect products and a lifting property (argument verification)

Let $G$ be a profinite group, and define the lifting property: $(*_p)$ For every extension $1 \to P \to E \to W \to 1$ where $E$ is finite and $P$ is a $p$-group and for every surjective morphism $f\...
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0answers
45 views

Let G be $\mathbb Z_p\times\dots\times \mathbb Z_p$. Find $A(G)$ (the Automorphism group of $G$). [duplicate]

Let G be $\underbrace{\mathbb Z_p\times\dots\times \mathbb Z_p}_{n \text{ times}}$. Find $A(G)$ (the Automorphism group of $G$). ($\mathbb Z_p$ is the integres modulo p, for example $\mathbb Z_2=\{0,...
3
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1answer
85 views

Let G be $\mathbb Z_p\times\dots\times \mathbb Z_p$ . Find A(G).

Let G be $\underbrace{\mathbb Z_p\times\dots\times \mathbb Z_p}_{n \text{ times}}$. Find $A(G)$. I know that $A(G)\cong GL_n(\mathbb Z_p)$. I prove it by taking $\varphi$ from $A(G)$ and show that ...
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0answers
34 views

Showing $d(\frac{Z_2(G)}{Z(G)})=d(G)d(Z(G))$ if and only if $d(G)=2$.

I am tring to proves following lemma Let $G$ be a finite p-group of coclass 3 and nilpotency class greater than 3 then $d(\frac{Z_2(G)}{Z(G)})=d(G)d(Z(G))$ if and only if $d(G)=2$ correct me ...
1
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1answer
47 views

If G is not abelian with order $p^3$ then $G'=Z(G)$

Question If $|G|=p^3$ and $G$ is not abelian show that $G'=Z(G)$ Attempt Since $|G|=p^3$ then $G$ is solvable and let $$1\leq G^{(n-1)}\leq...\leq G^{(n)}=G,(1)$$ be its derived series.We know ...
2
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1answer
49 views

Proof of theorem: If $G$ is a finite nonabelian $p$-group, then $\mathrm{Aut}_c(G)=\mathrm{Inn}(G)$ if and only if $G′=Z(G)$ and $Z(G)$ is cyclic.

Consider the following theorem: If $G$ is a finite nonabelian $p$-group, then $\operatorname{Aut}_c(G)=\operatorname{Inn}(G)$ if and only if $G′=Z(G)$ and $Z(G)$ is cyclic. Notation $p$ is a ...
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2answers
39 views

Show that the dihedral group $D_{16}$ is the internal direct product of its Sylow subgroups.

Show that the dihedral group $D_{16}$ is the internal direct product of its Sylow subgroups. (We use the notation $D_{16}$ for the dihedral group of order 32) Here's what I think. Since $D_{16}$ is ...
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0answers
15 views

Central quotient of $p$-groups

Suppose $P$ is a finite $p$-group with center $Z(P)$ of order $p$. What kind of groups can appear as the central quotient $P/Z(P)$? For example, the quotient is in particular a so-called capable ...
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0answers
19 views

Determine possible $p$-groups from center and quotient

Consider the following situation: I have given a finite $p$-group $P$ (in the case I am interested in $p = 2$) with cyclic center $Z(P)$ and I also know the structure of the quotient $P/Z(P)$ (which ...
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0answers
45 views

Subgroup of $p$-group and cyclic center [closed]

This is the problem: Let $p$ be a prime and $G$ a $p$-group. Prove that $Z(G)$ (center of $G$) is cyclic if and only if $G$ has a unique normal subgroup of order $p$. I can't see why having a unique ...
1
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1answer
39 views

Finite $2$-group with derived subgroup of order 8 [duplicate]

Does there exist a finite non-abelian $2$-group $G$ such that $G^{\prime}\cong D_8$ or $G^{\prime}\cong Q_8$? By an easy inspection with GAP, I could not find any example! Any answer or comment will ...
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2answers
83 views

Fundamental Theorem of Abelian Groups - intuition regarding Lemma

I'm learning group theory and have come across the Fundamental Theorem of Abelian Groups. My material is a book by Thomas W. hungerford called "Abstract Algebra: An Introduction". In this, Hungerford ...
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3answers
217 views

$p$-groups in which the centralizers are normal

Let $|G|=p^n$ and $p$ a prime and let $|G:C_G(x)|\leq p$ for all $x\in G$. Prove: $(a)~~~~~C_G(x)\trianglelefteq G$ for all $x\in G;$ $(b)~~~~~G’\leq Z(G);$ ${\color{red}{{(c)~~~~~|G’|\...
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1answer
70 views

Group of order $2^{67}$

Prove that a group of order $2^{67}$ has a normal subgroup of order $2^{59}$. I think we need to use the fact that it's a p-group, and then use the quotient group with the center here, but I am not ...
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1answer
42 views

Group of order 81 acting on a set of order 98

A group P acts on a set Ω. We know that|P| = 81 and |Ω| = 98. Let $Ω_0$ be the set of elements of Ω that are fixed by every element of P. In other words, $Ω_0$ ={α ∈ Ω | α · g = α for all g ∈ P}. Show ...
2
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0answers
67 views

Some conditions on a finite non-abelian $2$-group

Let $G$ be a finite non-abelian $2$-group, $\nu(G)$ denotes the number of conjugacy classes of non-normal subgroups of $G$ and $G^{\prime}$ denotes the derived subgroup of $G$. If $|G^{\prime}|=8$ ...
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0answers
26 views

Subgroup of index p in an infinite p-group?

Does an abelian infinite $p$-group always contain a subgroup of index $p$ ? Thanks.
2
votes
1answer
32 views

Let $P$ be a finite p-group, $A\vartriangleleft P$, with $|A|= p$. Show that $A \subseteq Z(P)$

Let $P$ be a finite p-group, $A\vartriangleleft P$, with $|A|= p$. Show that $A \subseteq Z(P)$ I have to show that if $x \in A \Rightarrow x \in Z(P)$. My attempt: Since A is a normal subgroup of ...
3
votes
2answers
83 views

If $G$ is a group of order $250,000 = 2^4 5^6$, show that $G$ is not simple.

If $G$ is a group of order $250,000 = 2^4 5^6$, show that $G$ is not simple. By the Sylow theorem, we have that the number of $2$-sylow subgroups of $G$ $n_2$ satisfy that $$ n_2 \equiv1\mod2\mbox{ ...
2
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0answers
68 views

cyclic subgroups of a $p$-group

Let $G$ be a finite non-Dedekind $p$-group and $\nu^*(G)$ denote the number of conjugacy classes of non-normal cyclic subgroups of $G$. Does there exist a normal second maximal subgroup $S$ of $...
2
votes
1answer
45 views

$G$ is a non-cyclic group of order $p^2$. Then $\langle g \rangle \cap \langle k \rangle = \{e\}$

I need to prove the following result: Let $G$ be a group of order $p^2$. Then $G \simeq \mathbb Z_{p^2}$ or $G \simeq \mathbb Z_p \times \mathbb Z_p$. and the results I have at my disposal are the ...
1
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1answer
80 views

Show that there is a nonabelian group of order $p^n$ with a cyclic subgroup of index $p$

Assume $p$ is a prime integer and let $n$ be an integer such that $n \geq 3$. Show that there is a nonabelian group of order $p^n$ with a cyclic subgroup of index $p$.
3
votes
1answer
81 views

Order of automorphism group of abelian group

In Derek Robinson's A Course in the Theory of Groups, exercise 1.5.13 states: Let $G=\mathbb{Z}_{p^{n_1}}\oplus\cdots\oplus\mathbb{Z}_{p^{n_k}}$, where $n_1<n_2<\cdots<n_k$. Prove there ...
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0answers
80 views

Does $p$ always divide $[N_G(H) : H]$ when $G$ is a $p$-group?

On this question, $G$ is a finite $p$-group and $H$ is a proper subgroup of $G$. The accepted answer says Note that p divides $|N_G[H]/H|$ so $N_G[H]/H$ has a subgroup of order $p$. But what ...
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1answer
41 views

Correspondence between finite Galois extensions and abelian p-groups

There is a correspondence between $k$-isomorphic finite separable normal extensions of $k$, $char(k) = p > 0$, and abelian p-groups. Put it formally, $$\forall \, G = Gal(\Bbb F/k), |G| < \...
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0answers
20 views

Min. number of generators of a finite $p$-group is $\dim_{\mathbb{Z}/p\mathbb{Z}}(G/G^p[G,G])$

Prove that the minimum number of generators of a finite $p$-group $G$ is $\dim_{\mathbb{Z}/p\mathbb{Z}}(G/G^p[G,G])$, the dimension of the vector space $\mathbb{Z}/p\mathbb{Z}$ over $\mathbb{Z}/p\...
2
votes
1answer
49 views

If $L_2/L_1$ is Galois, then $L_2 \cap K / L_1 \cap K$ is Galois

Suppose I have fields $F \leq L_1 < L_2 \leq E$, where $E/F$ is separable, $[E : F] = p^n$ for a prime number $p$, and $L_2 / L_1$ is Galois and of degree $p$. In addition, I have a subfield $F \...
1
vote
0answers
109 views

Subextension of a field with Galois series of subextensions of prime degree

Let $p$ be a prime number, and $E/F$ be a field extension. Suppose $E/F$ has a finite series of subfields $$ F = E_0 < E_1 < \cdots < E_n = E $$ with $E_i / E_{i-1}$ Galois of degree $p$ ...
3
votes
0answers
51 views

Intersection of subgroups with normal series of prime index

Let $G$ be a group, and $H$ a subgroup such that for a prime $p$, there is a normal series $$ H = H^0 \triangleleft H^1 \triangleleft \cdots \triangleleft H^n = G $$ between $H$ and $G$ such that ...