Consequences of equality among left and right cosets

Let G be a group, H \leq G a subgroup, and g \in G a fixed element. Prove that if Hg = kH for some element k \in G, then Hg = gH and g \in N_G(H).

Suppose Hg = kH for some k \in G. In particular, k \cdot 1 \in Hg; then k = hg for some h \in H, so that Hg = kH= hgH. Left multiplying by h^{-1} we have h^{-1}Hg = gH, but since h \in H, h^{-1}H = H, and we have Hg = gH. Moreover, g^{-1}Hg = H so that g \in N_G(H).

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  • Bobby Brown  On September 23, 2010 at 8:46 pm

    Why is k an element of Hg? I agree that if kH=gH, then k is an element of gH (or for right cosets), but why is this true for “mixing” left and right cosets?

    • nbloomf  On September 23, 2010 at 9:05 pm

      What is a coset?

      kH is the set of all products of the form kh where h \in H. Now let h = 1.

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