Proof of interesting equation: Related Topics, groups, semigroups, binary operations

Here is the question:

"Let G= Z x Q (Where Z is the set of integers and Q is the set of rational numbers) and define a binary operation o by

(a,b) o (c,d) = (a+c,2^{-c}b+d) - (*)

Is (G,o)

(i) a semigroup,

(ii) a group?

You must give full reasoning for your answer in order to get full marks."

[13 Marks]

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I require a natural proof to this and I am still a little new to this high level of maths so please excuse me for obvious errors.

I believe that I understand what the terms in the equation mean e.g. (binary operation - two elements with a calculation applied will produce an element in the same

set)

(semigroup - a set with associated binary operation, in this case o, that meets the condition a o (b o c) = (a o b) o c

(Group - a binary operation attatched to a set)

I don't understand how we are to apply the above principles to equation (*).

Any help to shed some light on this would be greatly appreciated.

Re: Proof of interesting equation: Related Topics, groups, semigroups, binary operati

Hey swcs.

For a group we have four axioms: 1) Closure 2) Identity 3) Inverse and 4) Associativity. Can you show these?