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