- #1
Bashyboy
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- 5
Homework Statement
Let ##G## be a group. I need to find a set ##X## and an injective function from ##G## into ##Sym(X)##
Homework Equations
The Attempt at a Solution
I am having difficulty with this problem, and I want to make sure I understand exactly what it is asking.
If I understand the question correctly, we have some group ##G##, and we want to find some set ##X## and create some mapping (specifically, an injection) which associates the elements in ##G## with the elements in ##Sym(X)##, so that we can clearly see that the elements of ##G##, along with its binary operator ##\star##, form functions?
Here are some ideas: I am pretty certain that I need to choose an ##X## such that ##Sym(X)## is at least as large as, if not greater than, ##G##. I know that the binary operator ##\star## is actually a function, so could I base my injection off of this? I have this suspicion that ##G=X## would work, but I can't quite figure it out.
EDIT: I also know that ##(Sym(X), \circ)## forms a group, so I am basically trying to find an injection between two different groups.
Does anyone have any suggestions?