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Is a factor group by a non-trivial normal subgroup is always smaller than the group ?

  1. Mar 23, 2012 #1

    Leb

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    This is not really a homework questions, rather a concept based one. I am studying from Fraleigh's ''Intro to abstract algebra'' and in chapter 15 it states, that for a group G and normal non-trivial subgroup of N of G, the factor group G/N will be smaller than G. I am not sure how he counts the change in size of the group.

    For instance(ex from wiki), if we take G=Z6 and it's normal subgroup N={0,3} we get G/N to be { {0, 3}, {1, 4}, {2, 5} } (i.e. all the cosets, which partition the whole of G). Or do we take each coset as a different element in G/N ?
    That is a={0,3}, b= {1,4}, c={2,5} so that |G/N| = 3 ?
     
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  3. Mar 23, 2012 #2

    HallsofIvy

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    Re: Is a factor group by a non-trivial normal subgroup is always smaller than the gro

    Yes, what you say is true. The elements of a factor group of group G are subsets of G. Further, they "partition" G- each element of G is in one and only one of these subsets. Finally, every coset contains the same number of elements. That is, if |G|= n and |N|= m then |G/N|= n/m.

    If m= 1, then N is just the identity and G/N= G. In all other cases, m> 1 so n/m< n.
     
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