At least how many normal subgroups of a group must have?

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Expert Solution

Every group has at least one normal subgroup, namely itself. The trivial group (the one that only has one element), only has that as a normal subgroup. All other groups have at least two normal subgroups, the trivial subgroup and itself.

Some groups only have those two normal subgroups. They’re called simple groups. Cyclic groups having a prime number of elements are examples. They’re the only Abelian groups with exactly two subgroups. Sometimes they’re excluded from simple groups since they differ a lot from the rest of the simple groups.

The smallest non-Abelian simple group is A5, the alternating group on five symbols. It has 60 elements.

Every group has at least one normal subgroup, namely itself

Nipun Maity answered 1 year ago

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