Paper 3, Section II, D

Groups | Part IA, 2018

Define the quotient group G/HG / H, where HH is a normal subgroup of a group GG. You should check that your definition is well-defined. Explain why, for GG finite, the greatest order of any element of G/HG / H is at most the greatest order of any element of GG.

Show that a subgroup HH of a group GG is normal if and only if there is a homomorphism from GG to some group whose kernel is HH.

A group is called metacyclic if it has a cyclic normal subgroup HH such that G/HG / H is cyclic. Show that every dihedral group is metacyclic.

Which groups of order 8 are metacyclic? Is A4A_{4} metacyclic? For which n5n \leqslant 5 is SnS_{n} metacyclic?

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