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- Inner_automorphism abstract "In abstract algebra an inner automorphism is a function in which an initial operation is applied, then another operation, and then the initial operation is reversed. With letters to indicate the operations and thing-being-transformed: f−1∘g∘f(X). Sometimes the initial action and its subsequent reversal change the overall result (\"raise umbrella, walk through rain, lower umbrella\" has a different result from just \"walk through rain\"), and sometimes they do not (\"take off left glove, take off right glove, put on left glove\" has the same effect as only \"take off right glove\").More formally an inner automorphism of a group G is a functionf : G → Gdefined for all x in G byf(x) = a−1xa,where a is a given fixed element of G, and where we deem the action of group elements to occur on the right.The operation a−1xa is called conjugation (see also conjugacy class), and it is often of interest to distinguish the cases where conjugation by one element leaves another element unchanged (as in the \"gloves\" analogy above) from cases where conjugation generates a new element (as in the \"umbrella\" analogy).In fact, saying that conjugation of x by a leaves x unchanged, i.e.a−1xa = x,is equivalent to saying that a and x commute:ax = xa. Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group.An automorphism of a group G is inner if and only if it extends to every group containing G.".
- Inner_automorphism wikiPageID "61839".
- Inner_automorphism wikiPageLength "8912".
- Inner_automorphism wikiPageOutDegree "44".
- Inner_automorphism wikiPageRevisionID "699775929".
- Inner_automorphism wikiPageWikiLink Abelian_group.
- Inner_automorphism wikiPageWikiLink Abstract_algebra.
- Inner_automorphism wikiPageWikiLink Adjoint_representation.
- Inner_automorphism wikiPageWikiLink Automorphism.
- Inner_automorphism wikiPageWikiLink Bijection.
- Inner_automorphism wikiPageWikiLink Category:Group_automorphisms.
- Inner_automorphism wikiPageWikiLink Category:Group_theory.
- Inner_automorphism wikiPageWikiLink Center_(group_theory).
- Inner_automorphism wikiPageWikiLink Centralizer_and_normalizer.
- Inner_automorphism wikiPageWikiLink Classical_group.
- Inner_automorphism wikiPageWikiLink Commutative_property.
- Inner_automorphism wikiPageWikiLink Complete_group.
- Inner_automorphism wikiPageWikiLink Conjugacy_class.
- Inner_automorphism wikiPageWikiLink Cyclic_group.
- Inner_automorphism wikiPageWikiLink Frattini_subgroup.
- Inner_automorphism wikiPageWikiLink Function_(mathematics).
- Inner_automorphism wikiPageWikiLink Function_composition.
- Inner_automorphism wikiPageWikiLink Group_(mathematics).
- Inner_automorphism wikiPageWikiLink Group_action.
- Inner_automorphism wikiPageWikiLink Group_homomorphism.
- Inner_automorphism wikiPageWikiLink Group_isomorphism.
- Inner_automorphism wikiPageWikiLink Identity_element.
- Inner_automorphism wikiPageWikiLink Identity_function.
- Inner_automorphism wikiPageWikiLink If_and_only_if.
- Inner_automorphism wikiPageWikiLink Isomorphism_theorem.
- Inner_automorphism wikiPageWikiLink Lie_algebra.
- Inner_automorphism wikiPageWikiLink Lie_group.
- Inner_automorphism wikiPageWikiLink Matrix_ring.
- Inner_automorphism wikiPageWikiLink Normal_subgroup.
- Inner_automorphism wikiPageWikiLink Outer_automorphism_group.
- Inner_automorphism wikiPageWikiLink P-group.
- Inner_automorphism wikiPageWikiLink Perfect_group.
- Inner_automorphism wikiPageWikiLink Powerful_p-group.
- Inner_automorphism wikiPageWikiLink Projective_line_over_a_ring.
- Inner_automorphism wikiPageWikiLink Quasisimple_group.
- Inner_automorphism wikiPageWikiLink Quotient_group.
- Inner_automorphism wikiPageWikiLink Regular_p-group.
- Inner_automorphism wikiPageWikiLink Ring_(mathematics).
- Inner_automorphism wikiPageWikiLink Ring_homomorphism.
- Inner_automorphism wikiPageWikiLink Ring_theory.
- Inner_automorphism wikiPageWikiLink Unit_(ring_theory).
- Inner_automorphism wikiPageWikiLinkText "Inner automorphism".
- Inner_automorphism wikiPageWikiLinkText "conjugate subgroup".
- Inner_automorphism wikiPageWikiLinkText "conjugate".
- Inner_automorphism wikiPageWikiLinkText "conjugation".
- Inner_automorphism wikiPageWikiLinkText "inner automorphism group".
- Inner_automorphism wikiPageWikiLinkText "inner automorphism".
- Inner_automorphism wikiPageWikiLinkText "inner".
- Inner_automorphism wikiPageWikiLinkText "sandwich operation".
- Inner_automorphism first "V.N.".
- Inner_automorphism id "I/i051230".
- Inner_automorphism last "Remeslennikov".
- Inner_automorphism title "Inner Automorphism".
- Inner_automorphism title "Inner automorphism".
- Inner_automorphism urlname "InnerAutomorphism".
- Inner_automorphism wikiPageUsesTemplate Template:Citation.
- Inner_automorphism wikiPageUsesTemplate Template:Clarify.
- Inner_automorphism wikiPageUsesTemplate Template:MathWorld.
- Inner_automorphism wikiPageUsesTemplate Template:Reflist.
- Inner_automorphism wikiPageUsesTemplate Template:Springer.
- Inner_automorphism wikiPageUsesTemplate Template:Sub.
- Inner_automorphism subject Category:Group_automorphisms.
- Inner_automorphism subject Category:Group_theory.
- Inner_automorphism hypernym Function.
- Inner_automorphism type Disease.
- Inner_automorphism type Function.
- Inner_automorphism comment "In abstract algebra an inner automorphism is a function in which an initial operation is applied, then another operation, and then the initial operation is reversed. With letters to indicate the operations and thing-being-transformed: f−1∘g∘f(X).".
- Inner_automorphism label "Inner automorphism".
- Inner_automorphism sameAs Q1139583.
- Inner_automorphism sameAs Automorphisme_intérieur.
- Inner_automorphism sameAs Automorfismo_interno.
- Inner_automorphism sameAs 内部自己同型.
- Inner_automorphism sameAs 내부자기동형사상.
- Inner_automorphism sameAs Inwendig_automorfisme.
- Inner_automorphism sameAs m.0gr2g.
- Inner_automorphism sameAs Inre_automorfi.
- Inner_automorphism sameAs Q1139583.
- Inner_automorphism sameAs 內自同構.
- Inner_automorphism wasDerivedFrom Inner_automorphism?oldid=699775929.
- Inner_automorphism isPrimaryTopicOf Inner_automorphism.