Answer to Question #105620 in Abstract Algebra for Sourav Mondal

Question #105620
1.If (G,*)is a group, then , * is the only binary operation defined on G

2.If every element of a group G has finite order, then G must be of finite order.
1
Expert's answer
2020-03-17T13:36:33-0400

1.and: Not necessary

Let "G=\\R" then "(G,+)" is a group.

But multiplication is also a binary operation on "G."

2.Ans: Not necessary

Counter example 1.

Let "G=\\frac{Q}{\\Z}" Then elements of "G" are looks like "\\frac{p}{q}+\\Z" ,whose order is "q" . But the order of the group is not finite.

Counter example 2.

Let "G=" "\\Z_2\u00d7\\Z_2\u00d7............" up to infinity time.

Then G is an infinite group but order of each elements is finite.


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