Misplaced Pages

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article needs attention from an expert in Mathematics. The specific problem is: Check if reference given is correct for this topic. WikiProject Mathematics may be able to help recruit an expert. (January 2025) This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Fully normalized subgroup" – news · newspapers · books · scholar · JSTOR (January 2025) (Learn how and when to remove this message)

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. Another way of putting this is that the natural embedding from the Weyl group of the subgroup to its automorphism group is surjective.

In symbols, a subgroup ${\displaystyle H}$ is fully normalized in ${\displaystyle G}$ if, given an automorphism ${\displaystyle \sigma }$ of ${\displaystyle H}$, there is a ${\displaystyle g\in G}$ such that the map ${\displaystyle x\mapsto gxg^{-1}}$, when restricted to ${\displaystyle H}$ is equal to ${\displaystyle \sigma }$.

Some facts:

• Every group can be embedded as a normal and fully normalized subgroup of a bigger group. A natural construction for this is the holomorph, which is its semidirect product with its automorphism group.
• A complete group is fully normalized in any bigger group in which it is embedded because every automorphism of it is inner.
• Every fully normalized subgroup has the automorphism extension property.

References

1. Kızmaz, M. Yası̇r (1 June 2024). "A generalization of Alperin Fusion theorem and its applications". sciencedirect.com. pp. 172–205. doi:10.1016/j.jalgebra.2024.02.016. Retrieved 14 January 2025.

This group theory-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Subgroup properties
• Group theory stubs