Homework Sourse1Help me solve pic problem steps by steps As much as possibl
1.Help me solve pic problem steps by steps. As much as possible.
2.Also add a questions or conjecture after it ( for example, could we have finite order elements in an infinite group?)
2. How many functions are there from Z4 to Z4? How many of the functions are group isomorphisms? Solution
the remainder of x mod 4. Find ker . To which familiar group is Z8/ ker isomorphic? Solution. Note that if x Z8, then (x) = 0 means that [x]4 = 0, so 4 must divide x. Therefore, the kernel of is precisely the set ker = {0, 4}. Since this subgroup has two elements, Z8/ ker must have order 4. Therefore, it is isomorphic to either Z4 or the Klein 4-group. We claim that it is Z4—note that the coset ker + 1 generates the quotient group: ker + 1 = {1, 5}(ker + 1) + (ker + 1) = ker + 2 = {2, 6} (ker + 2) + (ker + 1) = ker + 3 = {3, 7} (ker + 3) + (ker + 1) = ker + 4 = ker + 0 = {0, 4}, and these are all the cosets. Thus Z8/ ker is a cyclic group of order 4, so it is isomorphic to Z4
