If G={1,w,w^2} forms a group under multiplication then order of w isa) 1 b) 2 c) 3 d)4

1 Answer

tiburtius's profile pic

tiburtius | High School Teacher | (Level 2) Educator

Posted on

If `(G,cdot)` is multiplicative group then its order is defined as cardinal number of its set `|G|`. Since your group has 3 elements in its set (assuming `1 ne w ne w^2` ) the order of of group is 3.

Order of an element  `w` of a group is the smallest natural number `n` such that `w^n=e` where `e` is identity element. So if `1 ne w ne w^2`, then `o(w)=3`.

If `w^2=1` (`w` is its own inverse) then `o(w)=2` and if `w=1` (G is trivial group) then `o(w)=1`.

Note that order of an element cannot be greater than the order of the group because the group wouldn't be closed under multiplication hence it wouldn't be a group at all.