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futurebird
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I have the table for a non-albelian group. I know the subgroups of this group. I need to know which subgroups are normal. How can I tell?
A normal subgroup is a subgroup of a non-abelian group that is invariant under conjugation. This means that for any element in the original group, when it is conjugated with an element in the subgroup, the result will still be in the subgroup.
A subgroup is normal if and only if for every element in the original group, the left and right cosets of the subgroup are equal. This is denoted by H g = g H for all g in the original group and H in the subgroup.
Yes, a non-abelian group can have only one normal subgroup. This is the case for the alternating group An, where n > 4. The only normal subgroup of An is the subgroup consisting of the identity element, which is also the center of the group.
Normal subgroups play a crucial role in the construction of quotient groups. In fact, the quotient group is defined as the set of all left (or right) cosets of the normal subgroup in the original group, with a defined operation. This allows us to study the structure of the original group in a more manageable way.
Yes, there are a few special properties of normal subgroups in a non-abelian group. For example, the center of a non-abelian group is always a normal subgroup. Also, if a group has only two normal subgroups, it must be abelian.