# Group Theory : Conjugacy, Cayley Table, Subgroups and Quotient Groups

See the attached file.

Define the notion of conjugacy as it applies in a general group.

Prove that the inverses of a pair of conjugate elements are also conjugate.

Prove that conjugate elements have the same order. (6 marks)

The remainder of this question concerns the group G , whose Cayley table is as

[TABLE]

(b) Determine the inverse and the order of each of the elements of G.

(c) Simplify each of the following...

(d) Given that the only element conjugate to g is g itself (you need not prove this), determine the conjugacy classes of G.

(e) Find H , a normal subgroup of G having three elements. Identify the elements of the quotient group G/H , and determine its isomorphism type.

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#### Solution Summary

Conjugacy, cayley table, subgroups and quotient groups are investigated and discussed. The solution is detailed and well presented.