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Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if p is the lowest prime dividing the order of a finite group G , then any subgroup of index p (if such exists) is normal.
One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p 2 by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of exponent p for p > 2. Order p 4: The classification is complicated, and gets much harder as the exponent of p increases.
For example, the subgroup Z 7 of the non-abelian group of order 21 is normal (see List of small non-abelian groups and Frobenius group#Examples). An alternative proof of the result that a subgroup of index lowest prime p is normal, and other properties of subgroups of prime index are given in .
The reason is that for such a group G, the Frattini subgroup of G contains the commutator subgroup of G and hence the rank of G is equal to the rank of the abelianization of G. [14] The rank problem is undecidable for word hyperbolic groups. [15] The rank problem is decidable for torsion-free Kleinian groups. [16]
The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, / is abelian if and only if contains the commutator subgroup of . So in some sense it provides a measure of how far the group is from being abelian; the larger the ...
The subgroup of M 22 fixing a point in the 3-transitive 22-point permutation representation is sometimes called M 21, and is isomorphic to PSL(3,4) (i.e. isomorphic to A 2 (4)). A 4-transitive permutation group on 23 points, and is the point stabilizer of M 24 (in the 5-transitive 24-point permutation representation of M 24 ).
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Similarly, a normal subgroup N of G is said to be a maximal normal subgroup (or maximal proper normal subgroup) of G if N < G and there is no normal subgroup K of G such that N < K < G. We have the following theorem: Theorem: A normal subgroup N of a group G is a maximal normal subgroup if and only if the quotient G/N is simple.