# right/left coset calculation

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• Aug 31st 2010, 04:11 PM
demode
right/left coset calculation
The following is a worked problem:

http://img64.imageshack.us/img64/3213/34423202.gif

So, could anyone please show me how they've calculated (1,2,3)H and H(1,2,3)? I mean, for example for (1,2,3)H how did they get (1,2,3)(1,2)=(1,3)? Similarly for H(1,2,3), how (1,2)(1,2,3)=(2,3)?

Thank you.
• Aug 31st 2010, 09:34 PM
Isomorphism
\$\$ The notation (1,2,3) is a permutation function f with the property f(1) = 2, f(2) = 3, f(3) = 1.

\$\$ And the notation (1,2) is a permutation function g with the property g(1) = 2, g(2) = 1, g(3) = 3. (Whenever an integer is omitted, that means the function fixes that integer, lke 3 in the previous example)

\$\$ So (1,2,3)(1,2) is a notation f*g (composition of f and g). Can you calculate (1,2,3)(1,2) now?
• Sep 2nd 2010, 03:18 AM
demode
How about (1,2)(1,2,3)? Since there are two disjoint cycles, I said in the first cycle 1 goes to 2, and in the second cycle 2 goes to 3. Again in the first cycle 2 goes to 1, and in the second cycle 2 goes to 3. So we have:

(1,2)(1,2,3)=(3,2)

The correct answer should be (2,3) not (3,2). But I guess since this is a cycle of length two the order doesn't matter, does it? Is my method correct?

By the way I used the same method with your hint and I managed to get (1,2,3)(1,2)=(1,3).