# Tagged Questions

Use with the (group-theory) tag. Refers to questions concerning finite groups of prime power order or infinite p-groups such as PrÃ¼fer groups, pro-p-groups, and Tarski monsters. This tag is not for p-adic number systems.

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### When does prime $p$ divide a term in the numerator and denominator of ${p^m \cdot k}\choose {p^m}$ the same number of times?

This has questions comes to me via a proof to Sylow's First Theorem where $G$ is a finite group of order $p^m \cdot k$, the number $p$ is a prime divisor of $|G|$, and $p^m$ is the highest power of $p$...
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### What is $pA$ when $p$ is a prime number and $A$ an abelian group?

Let $A$ be a finite abelian $p$-group. I want to prove that $pA$ is also an abelian finite $p$-group, of order strictly less than the order of $A$. The problem is that I don't even know what does the ...
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### Extra Special p -group

How to prove that in an extra special group of order p^{1+2n},the order of any abelian subgroup is of at most p^{1+n}?
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### Image of p-group

Let $G$ be a group and $|G|=p^n$ for some prime $p$. If $f:G\to H$ is a surjective homomorphism, how do I know $H=f(G)$ also has cardinality a power of $p$?
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### Local group rings

Let $k$ be a field of characteristic $p$ and $G$ a finite group. How do you prove that if $kG$ is local then $G$ is a $p$-group? (I know how to prove the converse but not this implication).
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### The order of the normalizer of a $p$-subgroup of $S_{p}$ [closed]

I found it In Exercise in abstract algebra by Dummit and Foote. Let $P$ be a Sylow $p$-group of $S_p$. What is the order of $N_{S_p}(P)$?
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### Group of order $p^2$ [duplicate]

Let $G$ be a group of order $p^2$ where $p$ is a prime. I want to show that either $G$ is cyclic or is the product of two cyclic groups of order $p$. After some work, the problem reduces to showing ...
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### Proof of the fixed point theorem in group action.

$\textbf{Fixed point theorem:}$ Let $G$ be a $p$-group and let $S$ be a finite set on which $G$ operates. If the order of $S$ is not divisible by $p$, there is a fixed point for the operation of $G$ ...
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### Characteristic subgroups of non-abelian $p$-group

It is a difficult problem to determine all the characteristic subgroups of a non-abelian $p$-group. But, then, I would seek for as many characteristic subgroups as we can, for small order $p$-groups. ...
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### Automorphisms of a group and subgroup

Is there a finite $p$-group $G$ such that $G$ has less number of automorphisms than some subgroup: $$|\mbox{Aut}(G)|<|\mbox{Aut}(H)| \mbox{ for some }H\leq G.$$ If there is such a group, then can ...
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### Presentation for Special $p$-groups

Is a uniform presentation for special $p$ groups of rank 2 known? Thanks in advance.
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### Classify all groups of order $p^2q^2$ up to isomorphism

Let $p,q \in \mathbb{N}$ be prime numbers with the properties $2 < p < q$ and $q - 1 , q + 1 \notin \left\langle p \right\rangle$ Classify all groups of the order $p^2q^2$ up to isomorphism. ...
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### Homomorphisms $G\to\text{Aut}(G)$ for a $p$-group

I am interested in studying automorphisms of a $p$-group, or at least the highest power of $p$ dividing the order of the automorphism group, and feel like studying homomorphism $G\to\text{Aut}(G)$ ...
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### Highest power of $p$ dividing $|\text{Aut}(G)|$ if $|G| = p^\alpha$

What is known about the highest power of a prime dividing the number of automorphisms of a $p$-group? I know that it is at least $p$ provided $p^2$ divides $|G|$, but can we say more in some instances?...
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### What does “the subgroups of $G$ form a chain” mean?

I am being asked to show that: $G$ is a cyclic $p$-group $\iff$ its subgroups form a chain. What does "its subgroups form a chain" mean? Please keep in mind that I am just asking for the meaning of ...
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### Groups of order $64$ with abelian group of automorphism

G. A. Miller in 1913 constructed the first example of a non-abelian group of order $64$ with abelian group of automorphisms. It is the group G=(C_8\rtimes C_4)\rtimes C_2=\langle x,y,z\colon x^8, y^...
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### Frattini subgroup of a finite elementary abelian $p$-group is trivial

I would like to improve my proof of the following result: If $H$ is a finite, elementary abelian $p$-group, then $\Phi(H) = 1$. Here, $\Phi(H)$ is the Frattini subgroup, defined as the ...