Isoclinism of groups
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Contents
History
The notion of isoclinism seems to have been first introduced by Philip Hall mainly for the purpose of classifying finite p-groups, in his 1937 paper.
About this page
This page is mostly about the mappings that are used to define isoclinism. For more on the equivalence relation of being isoclinic, see isoclinic groups.
Definition
Short definition
An isoclinism is an isologism of groups with respect to the subvariety of abelian groups.
Full definition
For any group , let denote the inner automorphism group of , denote the derived subgroup of , and denote the center of .
Let denote the map from to defined by first taking the map given as and then observing that the map is constant on the cosets of .
An isoclinism of groups and is a pair where is an isomorphism of with and is an isomorphism of with such that . Explicitly, this means that for any , we have the following:
Pictorially, the following diagram must commute:
Two groups are said to be isoclinic groups if there is an isoclinism between them.
Definition in terms of homoclinism
An isoclinism is an invertible homoclinism of groups, i.e., a homoclinism for which both the component homomorphisms are isomorphisms. Equivalently, it is an isomorphism in the category of groups with homoclinisms.
References
Journal references
Original use
- The classification of prime-power groups by Philip Hall, Volume 69, (Year 1937): ^{Official link}^{More info}: Definition introduced on Page 133 (Page 4 within the paper)
Other uses
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Textbook references
Book | Page number | Chapter and section | Contextual information | View |
---|---|---|---|---|
Group Theory II (Grundlehren Der Mathematischen Wissenschaften 248) by Michio Suzuki. 10-digit ISBN 0387109161, 13-digit ISBN 978-0387109169^{More info} | 93 | Chapter 4 (Finite p-groups), Definition 4.28 | definition introduced explicitly, followed by facts about isoclinic groups |