Isoclinism of groups
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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.
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.
- The classification of prime-power groups by Philip Hall, Volume 69, (Year 1937): Official linkMore info: Definition introduced on Page 133 (Page 4 within the paper)
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|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 978-0387109169More info||93||Chapter 4 (Finite p-groups), Definition 4.28||definition introduced explicitly, followed by facts about isoclinic groups|