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Normal subgroup

  • Page ID
    19065
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    A subgroup H of a group G is normal in G (H \(\triangleleft\) G) if gH = Hg for any g ∈G. Equivalently, H ⊂ G is normal if and only if gHg-1= H for any g ∈G, i.e., if and only if each conjugacy class of G is either entirely inside H or entirely outside H. This is equivalent to say that H is invariant under all inner automorphisms of G.

    The property gH = Hg means that left and rights cosets of H in G coincide. From this one sees that the cosets form a group with the operation g1H * g2H = g1g2H which is called the factor group or quotient group of G by H, denoted by G/H.

    In the special case that a subgroup H has only two cosets in G (namely H and gH for some g not contained in H), the subgroup H is always normal in G.


    Normal subgroup is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Online Dictionary of Crystallography.

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