Abstract Algebra (Group)

Definition:

Let G be a nonempty set  together with binary operation. We say G is a group under this operation if the following properties are satisfied :

1.  Closure.

Any pair of elements can be combined without going outside the set.

2. Associativity.

The operation is associative, that is (ab)c=a(bc)  for all a,b,c  in G.

3. Identity.

There is an element e in G, such that ae=ea=a for all a in G.

4. Inverses.

For each element in G, there is an element b in G such that ab=ba=e.

Example:

Joseph A. Gallian book Page 51 no 1

Q:  Give two reasons why the set of odd integers under addition is not a group.

A: The first, addition is not closed operation in the set of odd integers.For instance: 5+3=8. Eight is an even number.

The second, it doesn’t have identity. Zero (the identity of addition) is not element of the set of odd integers.

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