Do you know this theorem?
A set H in a group G is a subgroup iff ?
Suppose that .
What can be said about ?
Let G be a group, and let
(That means H is the normal subgroup of G, right?)
a)Show that is also a subgroup of G.
My proof:
Suppose that x,y is in G, so that
Since H is normal in G, pick
Now,
Is this right?
b) If H is finite of order k, then is also finite of order k.
Let the above be the elements of .
Then,
Are all distint.
Because otherwise if,
This implies,
.
Which is an impossibility for distinct elements.
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Here is a useful theorem to know.
Theorem: Let be a subgroup of the following are equivalent.
1)For all and implies .
2)For all we have .
3)The left and right cosets coincide, i.e. .
All these 3 are equivalent formulations for "normal subgroup". My favorite is #1.