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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Let $G$ be finite $p$-group whose set of irreducible character degrees is $\{1,p^a\}$ and $H\unlhd G$ s.t. $G/H$ is nonabelian. Is $H\le Z(G)$? [closed]

Let $G$ be finite $p$-group whose set of irreducible character degrees is $\{1,p^a\}$ and $H$ is a normal subgroup of $G$ such that $G/H$ is a nonabelian group. May I assume that $H\le Z(G)$?
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$p$-groups containing normal subgroup of orders of powers of $p$. [closed]

Let $G$ be a $p$-group such that $|G| = p^e$ where $p$ is prime and $e \geq 1$. Prove by induction that $G$ contains a normal subgroup of every possible order $p^i$ for $i = 1,\ldots, e-1$. I am ...
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27 views

Is there a systematic way to find the center of a finitely-presented, finite $p$ group?

Given a finitely-presented and finite $p$-group $G = \langle S \mid R \rangle$, is there a systematic way to find its center? My end goal is to find a subnormal chain of every order $1, p, p^2, \dots, ...
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1answer
78 views

Showing $|G|=p^r \Rightarrow |G| \equiv |Z(G)| \pmod{p}$

If $|G|=p^r$, then $|G| \equiv |Z(G)| \pmod{p}$ I think it's enough to show that $\sum_{|o(x)>1|}|G:st(x)|=np, n\in\mathbb{N}$ and use the proposal $|G|=|Z(G)|+ \sum_{|o(x)>1|}|G:st(G)|, $ ...
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1answer
42 views

A question on normal noncyclic abelian subgroups

Let $P$ be a nonabelian $p$-group, where $p$ is an odd prime. Let $Q$ be a nonabelian normal subgroup of $P$. Does there exist a normal noncyclic abelian subgroup of $P$ that is contained in $Q$? ...
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1answer
49 views

Let $P$ be a $p$-subgroup of a finite group $G$. Prove that $[G:P]$ is congruent to $[N_G(P):P]\bmod p$.

I have the following question which is giving me a rather hard time. Let $P$ be a $p$-subgroup of a finite group $G$. By considering an appropriate action of $P$, prove that $[G:P]$ is congruent to $[...
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1answer
52 views

Let $G$ be a $p$-group. If every two maximal subgroups of $G$ are conjugated, then $G$ is cyclic

I already know: Let $|G|=p^{n}$. Every maximal group is normal, since two maximal subgroups are conjugated, then there is an unique $P$ normal subgroup of $G$ maximal with order $p^{n-1}$. Also, $G/P\...
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1answer
56 views

Quaternion group as a central extension of a $2$-group [closed]

I'll appreciate it if you help me to tackle this situation. I'm going to characterize 2-group $G$ whose two main properties such as $cd(G)=\{1,2\}$ and there is normal abelian subgroup $P$ such that $...
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Abelian divisible p-groups and Prüfer group

I've been asked to show tha if D is an abelian divisible p-group then is isomorphic to the sum of copies of $\mathbb{Z}(p^{\infty})$, which is to say that there exists a set $X$ such that $D\cong\sum_{...
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107 views

Definitions of solvable group

A solvable group seems to be variously defined as one with a composition series where all the composition factors are Abelian, or as one with a subnormal series where all the quotients are Abelian. ...
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58 views

Show that a group of order $175$ is not simple.

$|G| = 175 = 5^2 \times 7$ After small calculation I found that only possible value of $n_5 = 1$ and $n_7 = 1$. How to prove that $G$ is not simple group?
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170 views

How to find minimal generating sets efficiently?

Let $G$ be a group. A minimal generating set of $G$ is a subset of $R$ of $G$ that generates $G$ such that no proper subset of $R$ also generates $G$. For arbitrary groups is its not true (unlike ...
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42 views

Frattini subgroup of p-groups characteristics

"Assume $P$ is a $p$-group and $N$ is normal in $P$ with this property that $P/N$ is an abelian elementary group. Prove that $\Phi(P)$ is in $N$. (Note: $\Phi(P)$ is intersection of maximal ...
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1answer
91 views

Is there any function in GAP finding all maximal elementary abelian subgroup of a $p$-group $P$?

I know how to obtain all subgroups of a group in GAP. Is there any function in GAP (or algorithm that I could implemnt myself) for obtaining all maximal elementary abelian subgroups of a $p$-group $P$?...
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32 views

A doubt about a theorem in Khukhro's book

Some notations: For all group $G$ and $i\geqslant 1$ $\gamma_{i}(G)$ denotes the $i$-th term of the lower central series of $G$; $G^{i}=\langle x^{i};~x\in G\rangle$; For all set $X$, $S_X$ denotes ...
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26 views

Reference for “Every finite $p$-group can be expressed as a section of a powerful $p$-group”

I was reading about powerful $p$-groups. For an odd prime $p$, a finite $p$-group is called powerful if $[G,G] \subseteq G^p$, where $G^p = \langle g^p ~:~ g \in G \rangle$. and I found here that &...
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44 views

In a group $G$, of order $p^k$ has a subgroup of order $p^{k-1}$ prove this subgroup is normal. [duplicate]

I have looked at many proofs that show that there exists a normal subgroup of size $p^{k-1}$, but no proofs that all subgroups of size $p^{k-1}$ are normal.
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If $p$ is a prime then all the non trivial subgroups of $G$ with $\lvert G\rvert=p^2$ are cyclic

If $p$ is a prime then all the non trivial subgroups of $G$ with $\lvert G\rvert=p^2$ are cyclic. I tried looking online where does this result come from, but could not find any direct result. I ...
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2answers
66 views

“The p-power bounding” right multiplication in a finite group G under some special conditions

[Roughly speaking, the following question considers a special setting in which we want to prove a property in the form of $ord(g \sigma)\ |\ p^k$.] The Problem in Detail: Let $G$ be a finite group, ...
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1answer
72 views

Show that a group of order $2^2 \cdot 5 \cdot 23^r$ is solvable but not simple.

Let $G$ be a group of order $2^2 \cdot 5 \cdot 23^r$ where $r \gt 0$ a) Show that $G$ is not simple. b) Show that $G$ is solvable. My attempt: a) Assume that: $S$ is the set of sylow-$23$-subgroups ...
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3answers
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The number of groups of order 32

There are 51 groups of order ($32=2^5$). My question is how this number was computed. Graham Higman and Charles Sims gives an estimate for the number of $p$-goups (i.e. groups of order $p^n$ where $p$-...
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1answer
47 views

Why $\Psi(H) \le \exp(G)$ for any subgroup $H$ of a finite $p$-group $G$ where $\Psi(H)=\frac{\Sigma_{h\in H}|h|}{|H|}$?

I was reading this paper. I need help to understand a claim. Let's take $G$ which is a finite $p$-group. For any subgroup $H\le G$ we define $\Psi(H)$ to be the average order of the elements of $H$ ...
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1answer
46 views

Show an example of a finite abelian $p$-group that has $p^2 + p + 1$ subgroups of order $p$

Show an example of a finite abelian $p$-group that has $p^2 + p + 1$ subgroups of order $p$ My Path: I tried with the abelian group of $\mathbb{Z}_{63}=\mathbb{Z}_{9} \times \mathbb{Z}_{7} \times \...
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0answers
50 views

How to show that $G$ is nilpotent based on these conditions

Am trying to figure out a proof that if $G$ is a finite $p$-group, then $G$ is nilpotent ... Here, nilpotent means that a group $G$ is nilpotent if there exists a series $1 = H_0 \leq H_1 \leq ... \...
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1answer
71 views

How to choose ultraproducts to prove that the class of finite ($p$, torsion) groups is not elementary?

I want to use Łoś's theorem to show that finite groups, $p$-groups, and torsion groups do not form elementary classes. Thus, I have to construct the ultraproduct, say, of some finite groups that is ...
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30 views

Embedding finite $p$-groups in principal congruence subgroup

Let $p$ be a prime and $n, k$ positive integers. The group $GL_n(\mathbb{Z}/p^k \mathbb{Z})$ contains the principal congruence subgroup $GL_n(\mathbb{Z}/p^k \mathbb{Z})_1 = \{I_n + p M : M \in M_n( \...
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2answers
70 views

Is every direct product of (finite) cyclic groups abelian?

Long story short I'm halfway through a proof and have hit a step where I show that a group order $p^2$ is abelian. I did this by splitting into the $C_{p^2}$ case and the $C_p\times C_p$ case, the ...
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1answer
61 views

Let $p$ be a prime number and let $G$ be a $p$-group. Then $G$ contains a normal subgroup of order $p^k$ for every nonnegative $k\le r$

Let $p$ be a prime number, and let $G$ be a $p$-group: $|G|=p^r$ . Then $G$ contains a normal subgroup of order $p^k$ for every nonnegative $k\le r$ But are there any normal subgroup of order $p^n$ ...
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2answers
62 views

Show that a subset $X \subseteq G$ generates $G$ if and only if $X+pG$ generates $G$.

Let $p$ be a prime, and let $G$ be a finite abelian $p$-group. Define a subgroup $H<G$ to be maximal if there is no other subgroup $K<G$ such that $H<K<G$. Define the set $pG = \left\{ pg: ...
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54 views

Splitting theorems in finite p-groups

By a splitting theorem I mean a statement of the type: if $G$ has a normal subgroup $N$ and some hypotheses are satisfied, then there is a complement, that is a subgroup $Q$ of $G$ such that $G = N \...
4
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1answer
53 views

Automorphism group of a non-abelian group is divisible by $p^2$.

Suppose $G$ is a finite non-abelian group, so that $|G|=p^n$ for some positive integer $n\ge 3$ (where $p$ is prime). How can we prove that $|{\rm Aut}(G)|$ is divisible by $p^2$? We know that $G/Z(...
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1answer
64 views

How does the $p$-group fixed point theorem proof make sense?

Let $G$ be a finite $p$-group acting on a finite set $X$, and ${\rm Fix}_G(X)$ the subset of $X$ consisting of fixed points under this action. Then $$|X| \equiv | {\rm Fix}_G(X)| \pmod{p}.$$ The proof ...
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1answer
60 views

Understanding the proof of ${\rm ord}(G)=p^n \Rightarrow Z(G) \neq \{e\}$.

Let $p$ be a prime number and $G$ be a group. I have some troubles to understand the proof of ${\rm ord}(G)=p^n \Rightarrow Z(G) \neq \{e\}$. The proof is given as follow: Observe the Group $G$ acting ...
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1answer
31 views

Finite group having a $p$-subgroup has nontrivial center

Let $G$ be a finite group, $H\le G$ a $p$-subgroup. I want to show that $Z(G)$ is non-trivial. I let $G$ act on $H$ by conjugation - or vice versa - but I every time it ended up with $$Z(G)\subseteq Z(...
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1answer
29 views

Special groups and special linear groups

Is there a connection between a special group (i.e. a p-group with its derived group, center and frattini subgroup all equal) and the special linear group (i.e. group of matrices with determinant=1)? ...
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Showing $\Phi(G)=G'G^p$.

Prove $$\Phi(G)=G'G^p,$$ where $\Phi(G)$ is the Frattini subgroup of $G$, the intersection of all maximum subgroups of $G$, $G'=[G:G]$ is the commutator subgroup of $G$, and $G^p$ is the group ...
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1answer
43 views

Does the Prufer p-group contain direct sums of cyclic groups?

Let $G$ be the Prufer $p$-group. Then my question is, does $G$ contain subgroups which are isomorphic to a finite or countable direct sum of finite cyclic groups? Of course, some direct sums of ...
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1answer
29 views

When are elements of a p-group independent mod a power of p?

Suppose that an Abelian $p$-group $G$ can be written as a direct sum of countably many finite cyclic groups. Let $a_1,...,a_k$ be generators of $k$ distinct summands of this direct sum, and let these ...
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1answer
41 views

If $\theta :G\rightarrow H$ is a surjective homomorphism, then $\theta(\Phi(G))\leq\Phi(H)$.

This is a claim when I try to solve another problem related to the Fratinni group of a p-group, and I saw an answer Frattini subgroup of a finite elementary abelian $p$-group is trivial. I am stuck at ...
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1answer
60 views

If $G$ is group with $|G|=p^3$, $p$ prime, then $G'=Z(G)$

I'm trying to solve this problem from my abstract algebra text book: Being $G$ a non-abelian group of order $p^3$, with $p$ prime. Prove that $G'=Z(G)$ In my notation $G'$ is the derived subgroup of ...
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1answer
45 views

Does the direct sum decomposition of a p-group tell you the number of Prüfer subgroups?

My understanding is that every countable Abelian $p$-group can be written as a direct sum of at most countably many finite cyclic groups and at most countably many copies of the Prüfer $p$-group. ...
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32 views

When are two Abelian p-groups with the same character isomorphic?

Definition 2.6 of this journal paper defines a notion called “character” for Abelian $p$-groups: $\oplus_\alpha H$ denotes the direct sum of $\alpha$ copies of $H$ where $\alpha\leq\omega$. If $A=\...
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1answer
78 views

Prove that any group of order $945$ has at least one subgroup of order $9$

I'm trying to solve this group theory problem from my abstract algebra course. It goes like this: Prove that any group of order 945 has at least one subgroup of order 9. First, I noticed that $945=3^...
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2answers
84 views

Prove that any group of order $27$ is not simple

I'm stuck on this problem from my abstract algebra course: Prove that if $G$ is a group with $|G|=27$, then $G$ is not simple. First I noticed $|G|=27=3^3$. I thought I can use a statement I saw on ...
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46 views

Construction of an automorphism of elementary abelian p-group of order $p^{2}$

Let $G$ be an elementary abelian p-group of order $p^{2}$. Let $g$, $h$ be two non-trivial elements of $G$. I want to construct an automorphism $\phi\in Aut(G)$ such that $\phi(h)=g$, which is ...
4
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1answer
95 views

Prove that any group $G$ with $|G|=588$ is solvable

I'm stuck trying to solve this problem from my abstract algebra course: Prove that every group of order $588$ is solvable (If you assume that all groups of certain order are solvable, you must prove ...
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1answer
45 views

All 2-groups have cyclic commutator subgroups?

I have seen that groups of order 8 and order 16 have commutator subgroup cyclic. I want to know if that can be extended to all 2-groups or if there is a reason why this happens.
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1answer
57 views

How to prove the existence of non-abelian group of order $125$

The following question was part of my algebra assignment. Let $G$ be a non abelian group. Can its order be $125 ?$ $p$ groups have non-trivial center. So, if $|Z(G)| =125$ and the group is abelian, ...
5
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1answer
182 views

Conjugacy classes of a nonabelian group of order $p^4$

Apologies in advance for the long text! I will explain what I have done so far, and then I will present my question. Consider a nonabelian group of order $p^4$, where $p$ is a prime. Then, the center $...
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67 views

Bounds for probability that two elements commute in a group?

Let $G$ be a finite non-abelian group, and lets randomly choose two elements of $G.$ It seems quit well known that the probability that they commute is at most $\text{Pr}(G)\le\dfrac{5}{8}.$ Here is a ...

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