Suppose G is a group with center Z(G), inner automorphism group \operatorname{Inn} ... * Finite p-group with center of prime order and inner automorphism ... ...
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Suppose G is a group with finitely generated inner automorphism group ... this set under the quotient map. The inner automorphism that \sigma resembles ... ...
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Inner automorphism \to Automorphism
is a right tight function restriction ... G as a normal subgroup, and an inner automorphism \alpha of K such that ... ...
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order p. Suppose, further, that the inner automorphism group \operatorname{Inn} ... group of order eight, with the inner automorphism group a Klein four ... ...
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normal subgroup|group whose inner automorphism group is central in ... G is termed a normal subgroup whose inner automorphism group is central in ... ...
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G is trivial. Then, the fact about::inner automorphism group \operatorname{Inn} ... group \operatorname{Aut}(G) and inner automorphism group \operatorname{Inn} ... ...
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The fact about::inner automorphism group of an fact about::extraspecial group is a fact about::self-centralizing subgroup of the whole fact about::automorphism ... ...
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The fact about::inner automorphism group of a group need not be a ... group of order eight, with the inner automorphism group a Klein four ... ...
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A group whose inner automorphism group is central in automorphism ... A group H is a group whose inner automorphism group is central in ... ...
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An automorphism of a group is termed an inner automorphism if it can ... An automorphism \sigma of a group G is termed an inner automorphism ... ...
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Q: Why is an inner automorphism an automorphism? How is the group structure of a group related to that of its automorphism group?
A: This basically ... ...
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This is a list of common questions about inner automorphisms. See also inner automorphism (the main page on the subject).
Q: Why is an inner automorphism ... ...
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An automorphism \sigma of a group G is termed a locally inner automorphism ... | Weaker than::Inner automorphism || || || ||
|}
===Weaker properties=== ... ...
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The inner automorphism group of a group is defined in the following ... The inner automorphism group of a group G, denoted as Inn(G), is defined ... ...
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#redirect normal subgroup ... ...
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This page lists some offbeat, but correct, definitions of specific information about::inner automorphism.
==In terms of being able to extend to ... ...
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A subgroup H of a group G is termed a locally inner automorphism-balanced ... # Every inner automorphism of G restricts to a locally inner automorphism ... ...
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an automorphism \sigma of L a basic inner automorphism of L if \sigma = \exp ... We call an automorphism of L an inner automorphism if it can be expressed ... ...
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Q: Given an inner automorphism, is there a unique element for which this is the conjugation operation? If the element is not unique, is there ... ...
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subgroup and question about::inner automorphism|{{#ask: question about::normal subgroup|limit = 0|searchlabel = See more questions about normal ... ...
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G) by the defining ingredient::inner automorphism group \operatorname{Inn}(G). automorphism class is a coset of the inner automorphism group -- it is not ... ...
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stronger = inner automorphism|
weaker = IA-automorphism}} ... Any fact about::inner automorphism of a group is an fact about::IA ... ...
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is the same thing as an fact about::inner automorphism. ... ===Inner automorphism===
Inner automorphism
An automorphism \sigma ... ...
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The automorphism property of being an inner automorphism is stronger ... Any inner automorphism of a group is a class automorphism: it sends ... ...
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* Weaker than::Inner automorphism
* Weaker than::Normal automorphism: A normal automorphism is a weakly normal automorphism whose inverse is also ... ...
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By the fact that the centralizer of the inner automorphism group in the automorphism group is trivial for a centerless group, i.e., C_{\operatorname{Aut} ... ...
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* Weaker than::Inner automorphism
* Weaker than::Strong power automorphism
===Weaker properties===
* Stronger than::Monomial automorphism ... ...
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simple group, there exists an inner automorphism \phi of Aut(N) such that \phi \circ \rho = \rho \circ \sigma.
==Relation with other properties== ... ...
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Any pushforwardable automorphism is an inner automorphism. An automorphism \sigma of a group G is termed pushforwardable if, for any homomorphism ... ...
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* Weaker than::Inner automorphism
* Weaker than::Class-preserving automorphism
* Weaker than::Extended class-preserving automorphism ... ...
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Any inner automorphism of a subgroup lifts to an inner automorphism ... Let G \le H be groups and \sigma be an inner automorphism of G. Then ... ...
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Any inner automorphism of a group is an infinity-extensible automorphism. ... # uses::Inner is extensibility-stable: Any inner automorphism of a ... ...
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if it can be extended to an inner automorphism for every embedding ... simple group, there is an inner automorphism \phi of Aut(N) such ... ...
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Question: An inner automorphism sends every element to within its conjugacy class. Are there outer automorphisms that do so?
Answer: Yes. An automorphism ... ...
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* Weaker than::Inner automorphism
* Weaker than::Infinity-extensible automorphism
===Weaker properties===
* Stronger than::Finite-chain-extensible ... ...
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weaker = inner automorphism}}
==Statement== \sigma of G that is not an inner automorphism.
In fact, there exist ... ...
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Q: It is clear that an inner automorphism can be pulled back via any quotient map, i.e., an inner automorphism of a quotient arises from an inner ... ...
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class) is an defining ingredient::inner automorphism. need not be inner, although every inner automorphism is class-preserving ... ...
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Q: It is clear that an inner automorphism can be extended to any bigger group as an inner automorphism of that group -- simply use the same conjugating ... ...
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subgroup, as well as an fact about::inner automorphism \alpha of K, such that ... that arises as the restriction of an inner automorphism must be injective, ... ...
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