Question

Decide whether the following statements are True or False.

Justify the answer.
(a) Every G-set is also a group
(b) Let S be a G-set with s1,s2 in S and g in G. If gs1=gs2, then s1=s2
(b) Let S be a G-set with a group

Answer 1

(a) False, a G-set is not necessarily a group. (b) False, distinct elements can have the same image in a G-set.

Step-by-step explanation:

(a) The statement that every G-set is also a group is False. A G-set is a set equipped with an action of a group G on it, whereas a group is a set equipped with an operation that satisfies certain properties. While a G-set can have a group structure, it is not necessarily a group itself because it may not satisfy the properties needed for a group.

(b) The statement that if gs1=gs2, then s1=s2 is False. It is possible for distinct elements s1 and s2 in a G-set S to have the same image under an element g in the group G, so gs1=gs2. This is because the group action of G on S need not be injective.