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In abstract algebra, the center of a group ''G'', denoted ''Z''(''G''),〔The notation ''Z'' is from German ''Zentrum,'' meaning "center".〕 is the set of elements that commute with every element of ''G''. In set-builder notation, : The center is a subgroup of ''G'', which by definition is abelian (that is, commutative). As a subgroup, it is always normal, and indeed characteristic, but it need not be fully characteristic. The quotient group ''G'' / ''Z''(''G'') is isomorphic to the group of inner automorphisms of ''G''. A group ''G'' is abelian if and only if ''Z''(''G'') = ''G''. At the other extreme, a group is said to be centerless if ''Z''(''G'') is trivial, i.e. consists only of the identity element. The elements of the center are sometimes called central. ==As a subgroup== The center of ''G'' is always a subgroup of ''G''. In particular: #''Z''(''G'') contains ''e'', the identity element of ''G'', because ''eg'' = ''g'' = ''ge'' for all ''g'' ∈ G by definition of ''e'', so by definition of ''Z''(''G''), ''e'' ∈ ''Z''(''G''); #If ''x'' and ''y'' are in ''Z''(''G''), then (''xy'')''g'' = ''x''(''yg'') = ''x''(''gy'') = (''xg'')''y'' = (''gx'')''y'' = ''g''(''xy'') for each ''g'' ∈ ''G'', and so ''xy'' is in ''Z''(''G'') as well (i.e., ''Z''(''G'') exhibits closure); #If ''x'' is in ''Z''(''G''), then ''gx'' = ''xg'', and multiplying twice, once on the left and once on the right, by ''x''−1, gives ''x''−1''g'' = ''gx''−1 — so ''x''−1 ∈ ''Z''(''G''). Furthermore the center of ''G'' is always a normal subgroup of ''G'', as it is closed under conjugation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Center (group theory)」の詳細全文を読む スポンサード リンク
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