00:01
Hello, let's have a look at the question.
00:02
So here we have to show that any group of order 4 is either cyclic and hence isomorphic to z by 4z or isomorphic to z by 2 z into z by 2 z.
00:19
So now here let us take g is a group of order 4.
00:29
Now here if g has an element of order 4, then we can say that g is cyclic and hence we can say that it isomorphic to z by 4 z.
01:03
Now let us assume that there is no element of order 4.
01:22
So we can take let g is equals to e comma a, b and c, where e is the identity element.
01:41
Now by lagrange's theorem it must divide 4.
02:00
Hence we can say that oa can be 1, 2 or 4.
02:10
So if we take that oa is equal to 1, then a is equal to e which is not true.
02:22
Now here we can say that oa cannot be 4 by our assumption.
02:29
Hence the only option left for oa is equals to 2...