# group theory

## English

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### Noun

group theory (usually uncountable, plural group theories)

1. Used other than with a figurative or idiomatic meaning: see group,‎ theory; a theory about (particular kinds of) groups.
• 2007, Hedley G. Dimock, Rachel R. Kass, How to Observe Your Group, Captus Press, 4th Edition, page 3,
After you have read over the seven group theories you'll be clearer about what makes the most sense to you for the groups with which you want to use your observation skills.
• 2012, Stéphanie Delaune, Steve Kremer, Daniel Pasaila, Security Protocols, Constraint Systems, and Group Theories, Bernhard Gramlich, Dale Miller, Ulrike Sattler (editors), Automated Reasoning: 6th International Joint Conference, Springer, LNAI 7364, page 165,
In particular we study the case where cryptographic primitives are modelled using a group theory. Group theories are a special case of monoidal theories which have been extensively studied by F. Baader and W. Nutt [20,6] who have provided a complete survey of unification in these theories. Group theories include theories for exclusive or and Abelian groups.
2. (algebra) The mathematical theory of groups; (usually in combination) a branch of this theory.
• 1968, Arthur P. Cracknell (editor), Applied Group Theory: Selected Readings in Physics, Pergamon Press, page ix,
THE feeling seems to be developing among physicists that group theory can be used as and when required, as many other techniques already are, without its being necessary for everyone to study all the whys and wherefores of the mathematical details involved.
• 1995, Jean Brocas, Chapter 4: The Use of Group Theory in the Study of Non-Rigid Molecules, Danail Bonchev, D. H. Rouvray (editors), Chemical Group Theory: Techniques and Applications, Gordon and Breach, page 133,
Planar cis and/or trans internal rotation tunnelling have been discussed in terms of group theory, leading to theoretical spectra which have been compared to experimental work.
• 1999, Katalin Bencsath, et al., Aspects of the Study of Free Groups, Jean-Camille Birget, Stuart Margolis, John Meakin, Mark V. Sapir (editors), Algorithmic Problems in Groups and Semigroups, Springer, page 57,
Finally, as a different example, if ${\displaystyle {\mathfrak {C}}}$  is the category of topological spaces and continuous maps then the group theory is topological groups. A categorical free group in this context is a free group within one of these group theories — that is, a group for which there is a basis such that any morphism in the category can be extended to a group homomorphism.