|
Title: Show that every group of order 4 must be isomorphic to Z4 or Z2 x Z2? Post by: FISH0818 on Sep 28, 2012 Do I just have to show it for the cyclic and noncyclic case?
Title: Show that every group of order 4 must be isomorphic to Z4 or Z2 x Z2? Post by: tommyo0729 on Sep 28, 2012 Suppose G= {a, b, c, e} is a group of order 4 where e is the identity element.
Since G has order 4, a,b,c,e are distinct. If G is cyclic, then G is isomorphic to Z4. If G is not cyclic, consider an element say, a, which is not the identity element. the order of the element a must divide 4(order of the group). hence its order is either 1,2 or 4 if the order is 4 then 'a' is a generator of G thus G is cyclic(contradiction) if the order is 1 then 'a' is the identity element. (contradiction). Hence, the order of 'a' must be 2 Every non-identity element of G has order 2, hence G is isomorphic to Z2 x Z2 |