- #1
Bashyboy
- 1,421
- 5
Homework Statement
Let ##H## be a normal subgroup of a group ##(G, \star)##, and define ##G/H## as that set which contains all of the left cosets of ##H## in ##G##. Define the binary operator ##\hat{\star}## acting on the elements of ##G/H## as ##g H \hat{\star} g' K = (g \star g') H##.
Homework Equations
The Attempt at a Solution
I am having difficulty demonstrating that ##\hat{\star}## is well-defined. I understand that it is two cosets to be equal. So, for instance, we could have
##g_1 H = g_2 H##
and
##g_3 H = g_4 H##
So, to show that the operator is well-defined, I would have to show that
##g_1 H \hat{\star} g_3 H = g_2 H \hat{\star} g_4 H##
is true. However, I am having difficulty with this. If I understand correctly, ##g_1 H = g_2 H## does not necessarily imply that ##g_1 = g_2##.