quotient group

quotient group

English

Noun

quotient group (plural quotient groups)

1. (group theory) A group obtained from a larger group by aggregating elements via an equivalence relation that preserves group structure.
   • 1975, John R. Stallings, Quotients of the Powers of the Augmentation Ideal in a Group Ring, Lee Paul Neuwirth (editor), Knots, Groups, and 3-manifolds: Papers Dedicated to the Memory of R. H. Fox, Princeton University Press, page 101,

     This paper shows how to compute the quotient groups Jⁿ/Jⁿ⁺¹ (as well as the multiplicative structure of the graded ring consisting of these quotient groups).

   • 1983, David H. Sattinger, Branching in the Presence of Symmetry, Society for Industrial and Applied Mathematics, page 33,

     The Weyl group is the quotient group N_H/T_H, and in the present case the Weyl group is simply the permutation group S₃.

   • 2002, Alexander Arhangel'skii, Topological Invariants in Algebraic Environment, Miroslav Hušek, Jan van Mill (editors), Recent Progress in General Topology II, Elsevier (North-Holland), page 39,

     The class of reflexive groups doesn't behave nicely with regards to operations: a closed subgroup of a reflexive group need not be reflexive, and a quotient group of a reflexive group need not be reflexive.

Usage notes

For a normal subgroup N of G, the quotient group of N in G is denoted G/N (pronounced "G modulo N").

Synonyms

• (group obtained from a larger group by aggregating elements): factor group

Translations

group obtained from a larger group by aggregating elements

Finnish: tekijäryhmä Italian: gruppo quoziente m

Further reading

• Isomorphism theorems on Wikipedia.Wikipedia
• Normal subgroup on Wikipedia.Wikipedia
• List of group theory topics on Wikipedia.Wikipedia