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Groupprops β

Center-fixing automorphism-balanced subgroup

Definition

The following are equivalent definitions of center-fixing automorphism-balanced subgroup.

No. Shorthand A subgroup of a group is termed a center-fixing automorphism-balanced subgroup if ... A subgroup H of a group G is termed a center-fixing automorphism-balanced subgroup if ...
1 center-fixing automorphism-invariant, and center contained in group's center it is a center-fixing automorphism-invariant subgroup of the whole group, and its center is contained in the center of the whole group. H is invariant under any automorphism \sigma of G with the property that \sigma(x) = x for all x \in Z(G), and additionally, Z(H) \le Z(G). Here Z(H) and Z(G) denote respectively the centers of H and G.
2 center-fixing automorphism restricts to center-fixing automorphism any center-fixing automorphism of the whole group restricts to a center-fixing automorphism of the subgroup. For any automorphism \sigma of G with the property that \sigma(x) = x for all x \in Z(G), we have that \sigma(H) \subseteq H, and also that \sigma(x) = x for all x \in Z(H).

Relation with other properties

Formalisms

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Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
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Function restriction expression H is a center-fixing automorphism-balanced subgroup of G if ... This means that being center-fixing automorphism-balanced is ... Additional comments
center-fixing automorphism \to center-fixing automorphism every center-fixing automorphism of G restricts to a center-fixing automorphism of H the balanced subgroup property for center-fixing automorphisms Hence, it is a t.i. subgroup property, both transitive and identity-true