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Specifically, for a group ''G'' and a normal subgroup ''N'' of ''G'', there exists a bijection from the set of all subgroups ''A'' of ''G'' containing ''N'' onto the set of subgroups ''A′'' of ''G/N'' that maps a subgroup ''A'' of ''G'' to a subgroup ''A′'' = ''A/N'' of ''G/N''. For all ''A'',''B'' ≤ ''G'' containing ''N'', and subgroups of ''G/N'' ''A′'' = ''A/N'' and ''B′'' = ''B/N'', the following hold: # ''A'' ≤ ''B'' if and only if ''A′'' ≤ ''B′'', # if ''A'' ≤ ''B'', then the Index of ''A'' in ''B'' equals the index of ''A′'' in ''B′'', # <''A,B''>/''N'' = <''A′'',''B′''>, where <''A'',''B''> is the subgroup of ''G'' Generated by ''A'' ∪ ''B'', # (''A'' ∩ ''B'')/N = (''A′'') ∩ (''B′''), and # ''A'' is a normal subgroup in ''G'' if and only if ''A′'' is a normal subgroup in ''G′''. This list is far from inclusive. In fact, most properties of subgroups are preserved in their images under the bijection onto subgroups of a quotient group. SEE ALSO |
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