Homework Statement
Let a = (a1a2..ak) and b = (c1c2..ck) be disjoint cycles in Sn. Prove that ab = ba.
The Attempt at a Solution
Sn consists of the permutations of the elements of T where T = {1,2,3,...,n}
so assume we take an i from T. Then either i is in a, i is in b, or i is in...
Homework Statement
Prove that a k-cycle in the group Sn has order k.
Homework Equations
The Attempt at a Solution
I'm mostly confused on how to write this in math notation. I know it will have order k because a1 → a2 → a3 ... ak-1 → ak → a1 if we do the compositions K times. and so...
I'm meant to show the quotient group Z18/M is isomorphic to Z6.
The elements of Z18/M are
M + 0 = {0, 6, 12}
M + 1 = {1, 7, 13}
M + 2 = {2, 8, 14}
M + 3 = {3, 9, 14}
M + 4 = {4, 10, 15}
M + 5 = {5, 11, 16}
so, I need a function f that takes z18/m to Z6 , so the domain would be the...
Homework Statement
Show that Z18/M isomorphic to Z6 where m is the cyclic subgroup <6>
operation is addition
The Attempt at a Solution
M = <6> , so M = {6, 12, 0}
I figured I could show that Z18/M has 6 distinct right cosets if I wanted to do M + 0 = {6, 12, 0} M + 1 = {7, 13, 1}...