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Let $G$ be a group such that $g^{-2}hg^2=h$ for any $g,h\in G$. What is the name of the property mentioned above? It is weaker than requiring the group to be commutative.

By the way, if any one happens to know how to search for such properties on the internet, it would be great. I have asked a related question here but am not sure how it works with this question.

Zuriel
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    It's equivalent to saying every square is central, which is equivalent to the quotient group $G/Z(G)$ having exponent two. – anon May 31 '17 at 22:30
  • It is actually equivalent to commutativity. – Moishe Kohan May 31 '17 at 22:32
  • Thanks @arctictern! Do you happen to know any other properties and/or examples (preferably with references) about such groups? – Zuriel May 31 '17 at 22:32
  • @MoisheCohen, i do not see why this implies commutativity. Could you please show it? – Zuriel May 31 '17 at 22:33
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    @MoisheCohen I don't think that's true: in both the dihedral group of order $;8;$ and the quaternion group, the quotient $;G/Z(G);$ is $;2,-$ elementary...and in fact isomorphic with Klein's Viergruppe – DonAntonio May 31 '17 at 22:36
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    The only other property I know is that $G/Z(G)$ can't be cyclic, which rules out $G/Z(G)\cong \Bbb Z_2$. DonAntonio gave the two smallest examples. – anon May 31 '17 at 22:39
  • @arctictern Indeed so....but "cyclic non-trivial", though pedantic, is more accurate. – DonAntonio May 31 '17 at 22:44
  • Also equivalent to $g^{-1}hg = ghg^{-1}$. This is an interesting property but I'm not sure how often it comes up (outside of Abelian groups). – Trevor Gunn Jun 01 '17 at 00:17
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    Since having exponent $2$ implies abelian, this property implies nilpotent of class $2$. Moreover, I think it is conjectured that almost all finite groups have this property. (It is conjectured that almost all finite groups are central extensions of two elementary abelian $2$-groups, I think.) So it does "come up" quite often, in a sense :) – verret Jun 01 '17 at 01:57

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