# Tagged Questions

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### On finite 2-groups that whose center is not cyclic

Let $G$ be a finite 2-group such that $\left|\dfrac{G}{Z(G)}\right|=4$, $Z(G)$ is not cyclic and $Z(G)$ has at least one element of order 4. Then prove that there exists an automorphism $\alpha$ of ...
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### Commutators Calculus

I was trying to understand the above Corollary but I have a problem, namely why in the second to last line $A_0 \leq \zeta_p(G)$? Any ideas? Definitions By recurrence we define $[x,_0\, y]=x$; ...
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### Show finite group is $p$-group given some structure of group

Let $G$ be a finite group. If there exists an $a\in G$ not equal to the identity such that for all $x\in G$,$\phi(x) = axa^{-1}=x^{p+1}$ is an automorphism of $G$ then $G$ is a $p$-group. This is ...
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### Finite $p$-group in which all its maximal subgroups are cyclic

Let $G$ be a finite $p$-group, $|G|=p^n$. Let $M_1,\dots,M_r$ be all the maximal subgroups and suppose they are cyclic. Why is $\Phi(G)\le Z(G)$? $\Phi(G)$ is the Frattini subgroup. I have no idea ...
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### On $p$-groups with a unique minimal subgroup

If $G$ is a finite group with a unique minimal subgroup, we know that $|G|=p^n$. I have to prove that if $p\neq2$ then $G$ is cyclic. This is the contest. What I don't understand is the following ...
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### Cauchy's Theorem $\#$ of elements order $p$

In the stronger statement of Cauchy's Theorem it states the the number of elements of order $p$ is a multiple of $p$. http://en.wikipedia.org/wiki/Cauchy%27s_theorem_(group_theory)#Proof_2 I noticed ...
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### $|G|=p^3$ non abelian $\Longrightarrow\gamma_2(G)\lneq Z(G)$

Let $G$ be a non abelian group of order $p^3$. Hence $c=2$ ($c$ is the nilpotence class of $G$). I'll write down some notation, in order to be clear. Let $Z_0(G):=1$, $Z_1(G)=Z(G)$, $Z_{k+1}(G)$ ...
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### Classify $p$-groups in which all groups of the same order are isomorphic

The answer to “Are two subgroups of a finite $p$-group $G$, of the same order, isomorphic?” is definitely no. Such groups are very rare. How rare? Can you classify all finite $p$-groups $G$ such that ...
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### Are there asymptotically more nonabelian groups of order $p^k$ than there are abelian groups of order $\leq p^k$?

Let $\alpha(n)$ denote the number of isomorphism classes of abelian groups of order $n$ and $\alpha^\prime(n)=\sum_{m=1}^n\alpha(m)$. Similarly, define $f(p^k)$ to be the number of isomorphism ...
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### Extensions of elementary abelian p-groups by themselves.

How many distinct extensions of $C^k_p$ by $C^l_p$ are there where $C^n_p$ is elementary abelian of order $p^n$, p a prime?
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### Does every group whose order is a power of a prime p contain an element of order p?

I need to know if every group whose order is a power of a prime $p$ contains an element of order $p$? Should I proceed by picking an element $g$ of the group and proving that there is an element in ...
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### Understanding a proof about finite $p$-groups

I can't follow the reasoning of the author,in this proof: let $G$ be a finite $p-$group. If $H$ is a proper subgroup of G, then $H<N_G(H)$ (clearly $N_G(H)$ is the normalizer of $H$ and p is ...
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### There is no core free subgroup of order $p^2$ in a group of order $p^4$

By the classification of group of order $p^4$ ($p$ odd prime) from Burnside's book it seems to me that there is no core free subgroup of order $p^2$ in a group of order $p^4$. If I am not wrong there ...
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### Nilpotency class of normal product of dihedral 2-groups

Pessimistically paraphrasing Polya: “if Jack cannot answer a question, there is an easier question Jack also cannot answer.” Hence I ask: Given positive integers $a,b$ describe the set N_{a,b} = ...
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### Existence of a normal subgroup in a finite group.

Let $G$ be a finite group. If a Sylow $p$-subgroup $P$ of $G$ is contained in the centre, then does there exist a normal subgroup $N$ of $G$, such that $P \cap N = \{e\}$ and $PN=G$? Thanks in ...
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### Order of automorphism group of a $p$-group is divisible by $p$.

Suppose $G$ is a finite $p$-group (where $p$ is prime), so that $|G|=p^n$ for some positive integer $n\ge 2$. How can we prove that $|\text{Aut}(G)|$ is divisible by $p$? Here $\text{Aut}(G)$ ...
I would like to ask two questions. Kindly help me in this regard: (1) In Finite p-group with a cyclic frattini subgroup., user28083 has given finite $p$-group with cyclic frattini subgroup. I need ...