00:01
In this problem, our job is to find the group that isomorphic to the quotient group or factor group g mod h.
00:08
We're given that g is the group of units, mod 32, and h is the subgroup of g that consists of just 1 and 31.
00:18
So since g contains the elements of z32 that are relatively primed 32, those are the odd numbers from 1 up to 31.
00:30
So we can start listing the elements of g.
00:33
We have to list them all, but we certainly can.
00:38
And we see that the order of g is 16.
00:43
So we'll go ahead and list those elements here.
00:50
And now that we have that list, we can start calculating the elements of g mod h, which are cosets.
00:59
And we also note by lagrange's theorem that since we have 16 elements in g and 2 in h, we will have eight cosets and all eight distinct cosets.
01:09
So one element in g mod h, start listing those elements.
01:14
Here is the coset 1 times h, which will just give us 1 and 31.
01:25
Next we will do 3 times h.
01:28
So 3 times 1 is 1.
01:30
And notice here that 31 is negative 1 mod 32.
01:34
So we're going to get 3 and negative 3, which we can also write mod 32 as 29.
01:41
So these elements and cosets line up nicely as opposites of each other.
01:45
When we do 5h, we see that we get 5 and negative 5, or 27 mod 32.
01:58
7h will be 725.
02:05
And continuing, we get 9h, containing 9 and 23.
02:13
11h contains 11 and 21.
02:21
13h contains 13 and 19 and finally 15h contains 15 and 17.
02:30
15 and 17.
02:32
So 8 cosets in all, which is what we were expecting.
02:37
And so to determine the isomorphic structure of g mod h, let's look at orders of elements.
02:44
We know the order of element of the element 1h is 1 because that's the identity...