A group is considered abelian if its group operation is commutative, meaning the order in which elements are multiplied does not affect the result. In other words, for all elements a and b in the group, a * b = b * a.
A direct product of groups is a new group formed by combining the elements of two or more groups. The elements of the new group are ordered pairs, with the first element from the first group and the second element from the second group. The group operation is defined component-wise, meaning (a, b) * (c, d) = (a * c, b * d).
To prove that the direct product of two groups, G1 and G2, is abelian, you must show that for any elements (a, b) and (c, d) in the direct product, (a, b) * (c, d) = (c, d) * (a, b). This can be done by using the definition of the direct product and the properties of the individual groups G1 and G2.
Proving that a direct product of groups is abelian can provide valuable insight into the structure and properties of the individual groups G1 and G2. It can also be useful in solving problems and proving other theorems related to group theory.
Yes, the direct product of any number of abelian groups will also be abelian. This is because the group operation is commutative in each individual group, so it will also be commutative in the direct product. However, if at least one of the individual groups is not abelian, then the direct product will also not be abelian.