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What is a quotient group

  1. Jul 23, 2014 #1

    A quotient group or factor group is a group G/H derived from some group H and normal subgroup H.

    Its elements are the cosets of H in G, and its group operation is coset multiplication.

    Its order is the index of H in G, or order(G)/order(H).


    Extended explanation

    Coset multiplication of cosets g1H and g2H yields the coset g1g2H. Proof:

    Multiply every element of the two cosets together:
    [itex]\{ g_1 h_1 g_2 h_2 : h_1 , h_2 \in H \}[/itex]

    By self-conjugacy, we get
    [itex]\{ g_1 g_2 h_3 h_2 : h_3 , h_2 \in H \}[/itex]

    where each h3 need not equal the h1 it was derived from. By closure of H, we get
    [itex]\{ g_1 g_2 h : h \in H \}[/itex]

    or the coset g1g2H.

    There are two trivial cases:
    H is identity group -> G/H is isomorphic to G
    H = G -> G/H is the identity group

    The simplest nontrivial case is for where H has half the number of elements of G. It has one coset, G - H, which is both a left and a right coset, making H a normal subgroup for every possible H with that order. Its coset multiplication table is
    H * H = H
    H * (G-H) = (G-H)
    (G-H) * H = (G-H)
    (G-H) * (G-H) = H

    This shows that G/H is Z(2), the 2-element cyclic group.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
  2. jcsd
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