is normal if its left and right cosets coincide i.e.
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A group is a simple group if it is nontrivial and has no proper nontrivial normal subgroups.
- That is, it does not have a factor group that is not the trivial subgroup or the group itself.
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(Fraleigh 14.13) The following are equivalent conditions for
to be normal. and , the following holds is invariant under all inner automorphisms of . That is
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(Fraleigh e14.31) Let
, then the intersection is also normal -
(Fraleigh e14.34) If
has exactly one subgroup of a given order, then is a normal subgroup of . -
A maximal normal subgroup of a group
is a normal subgroup such that there is no proper normal subgroup -
(Fraleigh 15.18)
is the maximal normal Subgroup of if and only if is simple.