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This is probably standard for group-theorists: Let $G$ be a finite group. Is it true that the intersection of all normalizers of subgroups equals the center? If so, where do I find a proof? What about the same question for infinite groups?

The original question can be reformulated as follows: Let $G$ be a finite group and fix $g\in G$. Assume that for every $x\in G$ there exists a natural number $k(x)$ such that $$ gxg^{-1}=x^{k(x)}, $$ does it follow that $k(x)=1$ for all $x$? One gets the reformulation by applying the original statement to cyclic groups. It suffices to consider cyclic groups, as an element that normalizes all cyclic groups, normalizes every subgroup.

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4 Answers

up vote 10 down vote accepted

No. There are non-abelian groups $G$ for which all subgroups are normal, such as the quaternion group of order 8. So the intersection of all normalizers is just $G$.

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This should generalize easily to infinite groups, right? Just take a direct sum of an infinite number of quaternion groups... –  Nick Gill Apr 18 '13 at 10:13
    
It is not obvious, whether the product of two Hamiltonian groups is Hamiltonian. In fact it is wrong, according to the classification in the wikipedia article. –  Jan Weidner Apr 18 '13 at 10:35
    
@Jan, good point! The $C_4\times C_4$ subgroups of $Q\times Q$ aren't all normal. In any case, the classification given in the wikipedia article includes some infinite groups, so the question is answered. –  Nick Gill Apr 18 '13 at 10:49
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The intersection of all normalizers of subgroups of a group $G$ is called the norm of $G$. By a result of Schenkman [E. Schenkman, On the norm of a group, Illinois J. Math., 7 (1960) 150-152] the norm of a group always lies in $Z_2(G)$ of the upper central series. So the question has positive answer if for example the center $Z(G)=Z_2(G)$.

The Schenkman's result has been improved by Cooper [C.D.H. Cooper, Power automorphisms of a group, Math. Z., 107 (1968) 335-356.] as follows: an automorphism which leaves every subgroup of a group $G$ invariant induces the trivial automorphism in the central factor group $G/Z(G)$.

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In particular, if norm of G is nontrivial, the center of G is nontrivial as well. –  Misha Apr 18 '13 at 14:22
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A Hamiltonian group is a non abelian group, in which every subgroup is normal. For example the Quaternion group $$\{1,i,j,k,-1,-i,-j,-k\}$$ is Hamiltonian. This gives a counterexample to your first question.

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As was pointed out by Rickard and Weidner, the existence of Hamiltonian groups like $Q_8$ shows that the answer to the original question is "no". But all Hamiltonian groups have even order, so this leaves open the question for odd order groups. The answer is "no" for these groups too.

Let $p$ be an odd prime, and let $G$ be the unique nonabelian group of order $p^3$ and exponent $p^2$. Then $Z(G)$ has order $p$, but $G$ has an elementary abelian subgroup $E$ of order $p^2$, consisting of all elements $x \in G$ with $x^p = 1$. I claim that $E$ normalizes every subgroup of $G$. To see this, let $H \subseteq G$. If $|H| \geq p^2$ then $H$ is normal in the whole group $G$, and otherwise $H \subseteq E$, and since $E$ is abelian, $E$ normalizes $H$.

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