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If H and K are subgroups of G, and K is normal, then HK is a subgroup of G.

  1. Jul 5, 2012 #1
    Ignoring the fact that it is redundant at times, is this proof correct? Also, is there a way to show that same result using the fact that K is closed with respect to conjugates rather than the fact that for all a in G, aK=Ka. Thank you!! :)





    Proposition: If H and K are subgroups of G, and K is normal, then HK is a subgroup of G.
    Proof: Let H,K be subgroups of G with K normal. We wish to prove that HK is a subgroup of G.

    Closure
    Take any two elements of HK, say hk and h’k’. Then, we want to show that (hk)(h’k’) is in HK. We note that K is normal, which implies that for kh’, there is some k’’ in K such that kh’=h’k’’ since for all a in G, aK=Ka. Thus,
    (hk)(h’k)’=h(kh’)k’=h(h’k’’)k’=hh’k’’k’. Then since hh’ is in H and k’’k’ is in K, we have that hkh’k’ is in HK.

    Identity
    Since H and K are both subgroups and therefore contain e, we have that HK must also contain e.

    Inverses
    Take any element hk in HK. We wish to show that (hk)-1 is in HK. We note that (hk)-1=k-1h-1. Then since K is normal and for any a in G, we have that aK=Ka, there exists some k’ in K such that k-1h-1= h-1k’. And, we have that h-1k’ is in HK. Hence, k-1h-1 is HK and every element in HK has an inverse in HK.


    Therefore, we conclude that HK is a subgroup of G.

    QED
     
  2. jcsd
  3. Jul 5, 2012 #2

    micromass

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    That seems alright.

    If you want to work with conjugates, then perhaps doing something like follows could work:

    [tex](hk)(h^\prime k^\prime)=hh^\prime [(h^\prime)^{-1} k h^\prime] k^\prime[/tex]

    and use that K is closed under conjugates.
     
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