Question

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.

Answer 1

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 $G=$ $\Z_2×\Z_2×............$ up to infinity time.

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