CENTER (GROUP THEORY)

In abstract algebra, the 'center' of a group ''G'' is the set ''Z''(''G'') of all elements in ''G'' which commute with all the elements of ''G''. That is,
:$Z(G)\; =\; \{z\; in\; G\; |\; gz\; =\; zg\; ;mbox\{for\; all\},\; g\; in\; G\}$.
Note that ''Z''(''G'') is a subgroup of ''G'', because
# ''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'').
Moreover, ''Z''(''G'') is an abelian subgroup of ''G'', a normal subgroup of ''G'', and even a strictly characteristic subgroup of ''G'', but not always fully characteristic.
The center of ''G'' is all of ''G'' if and only if ''G'' is an abelian group. At the other extreme, a group is said to be 'centerless' if ''Z''(''G'') is trivial, i.e. consists only of the identity element.
Consider the map ''f'': ''G'' → Aut(''G'') from ''G'' to the automorphism group of ''G'' defined by ''f''(''g'') = φ_''g'', where φ_''g'' is the automorphism of ''G'' defined by φ_''g''(''h'') = ''ghg''⁻¹. This is a group homomorphism, and its kernel is precisely the center of ''G'', and its image is called the inner automorphism group of ''G'', denoted Inn(''G''). By the first isomorphism theorem we get
:$G/Z(G)cong$
m{Inn}(G).

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Examples

See also

Examples

★ The center of the group $mbox\{GL\}\_n(F)$ of ''n''-by-''n'' invertible matrices over the field $F$ is the collection of scalar matrices $\{\; sI\_n\; |\; s\; in\; Fsetminus\{0\}\; \}$.

★ The center of the orthogonal group $O(n,\; F)$ is $\{\; I\_n,-I\_n\; \}$.

★ The center of the quaternion group $Q\; =\; \{1,\; -1,\; i,\; -i,\; j,\; -j,\; k,\; -k\}$ is $\{1,\; -1\}$.

★ The center of the multiplicative group of non-zero quaternions is the multiplicative group of non-zero real numbers.

★ Using the class equation one can prove that the center of any non-trivial finite p-group is non-trivial.

See also

★ center (algebra)

★ centralizer and normalizer

★ conjugacy class.