quotient group

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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 NH/TH, 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[edit]

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


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


Further reading[edit]