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So I was working through some problems in a book on $p$-groups and noticed that $p$-groups have some really nice properties. So I started computing what the values of $|G/Z(G)|$ for $p$-groups. I decided to see what it would be and found that it cannot attain the value of $p$. I am interested in the whethere $|G/Z(G)|$ can attain all other powers of $p$?

In general I am interesting in the values that $|G/Z(G)|$ can achieve.

Any help is greatly appreciated.

Pax
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    For example, any $n$ such that every group of order $n$ is cyclic (e.g. the $n$ such that $(n,\varphi(n))=1$) are not attainable. For then, $G/Z(G)$ would necessarily be cyclic, so that $G$ is abelian so that $|G/Z(G)|=1$ . I don't know if this is the only restriction though... This is if I am understanding your question correctly: characterize $n$ which occur as $|G/Z(G)|$ for some group $G$. – Alex Youcis Oct 05 '13 at 00:10
  • PS (+1), interesting question! Also, the e.g. in my last question should be i.e. The numbers $n$ such that every group of order $n$ is cyclic are precisely the $n$ such that $(n,\varphi(n))=1$. – Alex Youcis Oct 05 '13 at 00:11
  • Okay, so the primes are out immediately. That is indeed my question =) – Pax Oct 05 '13 at 00:18
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    The primes, and more generally products $p_1\ldots p_m$ such that $p_i\not\equiv 1\mod p_j$ for all $i,j$. Also, I'm a little confused by the indexing of your sets $\psi_n$ though. I see no dependence on $n$. – Alex Youcis Oct 05 '13 at 00:19
  • Sorry, I meant to break it up to try to solve a slightly more general problem, but I forgot. Thanks for reminding! – Pax Oct 05 '13 at 00:26
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    Here are some comments, but not a complete solution. First, if $n=p_1^{\alpha_1}\cdots$ and all the $\alpha_i>1$, your results on $p$-groups imply $n\in\psi$. Second, if $n$ is even - say $n=2m$ - then the dihedral group $D_{2n}$ is such that $D_{2n}/Z(D_{2n})$ has order $n$. So all even numbers are in $\psi$. The odd $n$ case is much harder. –  Oct 07 '13 at 07:37
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    There are certain numbers that cannot be in $\psi$: as already remarked, the cyclic numbers; also, abelian numbers with one or more of the $\alpha_i=1$ (as an example, $45\notin\psi$); there are other cases, but they seem harder to classify. Note that the much harder problem of finding groups that are central quotients of other groups is the classification of capable groups problem, of which there is much literature (a lot by our our own @ArturoMagidin). –  Oct 07 '13 at 07:37
  • @SteveD Why can't we have abelian numbers with one or more of the $\alpha_i=1$? We know we can't have $G/Z(G)$ be cyclic, but what does having it be abelian with a cyclic factor tell us? Sorry if this is a stupid question.:) – Alex Youcis Oct 09 '13 at 23:58
  • @AlexYoucis: Not stupid at all! It comes from an article of Baer, which essentially initiated the study of capable groups: Groups with Preassigned Center and Central Quotient. It is the Corollary to his Existence Theorem that is of most concern here. –  Oct 10 '13 at 02:13
  • @SteveD Ah, interesting! Thanks, I'll have to check that out. – Alex Youcis Oct 10 '13 at 02:14

2 Answers2

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The result you found that $|G/Z(G)|$ cannot be prime is essentially the only restriction for $p$-groups:

For every $k\geq 2$ and every $p\in\mathbb{P}$ there exists a finite $p$-group $G$ such that $|G/Z(G)|=p^k$.

Proof: Let $q:=p^{k-1}$ and consider the matrix group $\begin{pmatrix}1&\mathbb{F}_p&\mathbb{F}_q \\ &1&\mathbb{F}_q \\ && 1\end{pmatrix}$ which has order $p^{2k-1}$ and center of order $p^{k-1}$ (the top right corner) which gives us a center quoient of order $p^k$ as desired. QED.

Observe that by taking direct products of such groups one can easily write down examples of $|G/Z(G)|=n$ for every natural number $n>1$ in which every prime factor that occurs, occurs at least twice. This is because $Z(G_1\times G_2)=Z(G_1)\times Z(G_2)$.

Johannes Hahn
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In connection of this question it might be interesting to note that bounding the size of $G/Z(G)$ for a given finite group $G$ has some interesting history. For a given finite group Jordan's theorem gives an upper bound on the size of $G/Z(G)$ in terms of the degree of the smallest faithful representation of $G$.

user26857
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Felix
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