# Class equation

The class equation of a group G is 1+4+5+5+5.

a) Does G have a subgroup of order 5? If it does, is it a normal subgroup?

b) Does G have a subgroup of order 4? If it does, is it a normal subgroup?

c) Determine the possible class equations of nonabelian groups of order 8 and of order 21.

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#### Solution Preview

The order of a group is 20= .

a) Silow theorem implies that it has a subgroup of the order 5. Since 5 is prime, this subgroup is cyclic, and its any non-trivial element is it's generator. Then the order of its conjugacy class is a divisor of =4. Therefore, it belongs to the class of the order 4. Hence coinsides with the subgroup {1} , where C is the conjugacy class containing exactly 4 elements. This subgroup is obviously normal, since it satisfies the property: if intersect a conjugacy class, then contains this class.

b) Silow theorem implies that it has a subgroup of the order 4= . The order of it's any element is either 4 or 2. If the order is 4, then its conjugacy class is a divisor of =5. Then it belongs to a conjugacy ...

#### Solution Summary

Class equations are assessed in this solution.