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What is the relationship between abelian subgroup of a non-abelian group(when exist, example, theorem)?? any thing such link regarding the question would help.

 I apologize for previous post-error.
Michael
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    How can you lose commutivity? Did you perhaps mean the abelian subgroup of a non-abelian group? – Joffan Feb 09 '15 at 17:53
  • You can have abelian subgroups in a non-abelian group, but not non-abelian subgroups in an abelian group. – Chris Leary Feb 09 '15 at 17:56
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    Another way of saying that it is not possible is that the class of abelian groups forms a variety, and therefore is closed under taking subalgebras. – lisyarus Feb 09 '15 at 18:00

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If $H$ is a subgroup of an abelian group, and $x,y \in H$ then $x,y \in G$ and hence $$xy=yx$$

N. S.
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It is the other way around. I mean, it is possible for a subgroup of noncommutative group to be commutative: just take any nonabelian group $G$ (for example $S_5 \times \mathbb Z_{24}$) and look at its center, the set of elements that commute with everything, $Z(G) := \{g \in G : (\forall h \in G)(hg=gh)\}$ (in this example: $Z(G) = \{e\} \times Z_{24}$).