This is a consequence of the first isomorphism theorem, because is precisely the set of those elements of that give the identity mapping as corresponding inner automorphism ( conjugation changes nothing ).
12.
Thus the elements and exchange places and, in addition, is twisted by the inner automorphism corresponding to this ensures that the product of the components of remains the identity element.
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In all these exceptional cases the subalgebra is the fixed point algebra of an inner automorphism of period 3, except for E 8 / A 4 �A 4 where the automorphism has period 5.
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For " n " even there is instead an outer automorphism interchanging the two types of reflections ( properly, a class of outer automorphisms, which are all conjugate by an inner automorphism ).
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If the inner automorphism group is trivial ( when a group is abelian ), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group.
16.
Note that the specific quotient set depends on a choice of maximal torus, but the resulting groups are all isomorphic ( by an inner automorphism of " G " ), since maximal tori are conjugate.
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The graph automorphism is unlikely to be an inner automorphism, so probably they are not conjugate within the group itself, but they would be in the holomorph . talk ) 00 : 34, 24 August 2009 ( UTC)
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In mathematics, in the field of group theory, a subgroup of a group is said to be "'fully normalized "'if every automorphism of the subgroup lifts to an inner automorphism of the whole group.
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If \ phi = \ phi _ a is an inner automorphism ( conjugation by some element " a " ), then it acts trivially on representations, because representations are class functions ( conjugation does not change their value ).
20.
For, except for, the automorphism group of A " n " is the symmetric group S " n ", with inner automorphism group A " n " and outer automorphism group Z 2; the outer automorphism comes from conjugation by an odd permutation.
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