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Math Help - Index of 2 is equivalent to normal

  1. #1
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    Index of 2 is equivalent to normal

    Hello,

    I have a homework problem which asks:
    Let G be a group and let H be a subgroup of G.
    (a) Show that if H is index 2 in G then H is a normal subgroup of G.
    (b) Show that if H is a normal subgroup of G then H is index 2 in G.

    I proved the first part like so:
    Assume H has an index of 2
    Thus it has 2 left cosets which are forced to be H and xH = G/H
    and the 2 right cosets are H and Hx = G/H
    So, Hx = xH for all x
    Thus H is normal.

    However, I can't figure out how to go the other way.

    Any help would be much appreciated.
    Thanks.
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  2. #2
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    Quote Originally Posted by epsilon View Post
    Hello,

    I have a homework problem which asks:
    Let G be a group and let H be a subgroup of G.
    (a) Show that if H is index 2 in G then H is a normal subgroup of G.
    (b) Show that if H is a normal subgroup of G then H is index 2 in G.

    I proved the first part like so:
    Assume H has an index of 2
    Thus it has 2 left cosets which are forced to be H and xH = G/H
    and the 2 right cosets are H and Hx = G/H
    So, Hx = xH for all x
    Thus H is normal.
    .
    (a)Consider H and H* (means H complement). We need to show xH = xH. If x in H then xH = H = hX. If x not in H then xH = H* = Hx.

    (b)Garbage! Take H = { e } and G = Z_3. So?
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