(a) Let H be a subgroup of a group G and let a be in G. Prove that (a^(-1))Ha={(a^-1)ha l h in H} is a subgroup of G that is isomorphic to H.
(b) If H is finite, prove that lHl=l(a^-1)Hal.
(b) If H is finite, prove that lHl=l(a^-1)Hal.
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(a) Let K = a^-1Ha. Clearly e is in K since e is in H, a subgroup.
Let s and t be in K. then s = a^-1 x a for some x in H, and t = a^-1 y a for some y in H.
so st = (a^-1 x a) ( a^-1 y a) = a^-1 xy a is in K since xy is in H.
And s^-1 = a^-1 x^-1 a is in K since x^-1 is in H.
So H is a subgroup.
A function f: G-->G where f(g) = a^-1 g a is known as an inner automorphism. The second part of (a) is to show that an inner automorphism takes a subgroup H to an isomorphic image of H = a^-1 H a.
Clearly f is 1-1 since if a^-1 s a = a^-1 t a then
a(a^-1 s a)a^-1 = a(a^-1 t a)a^-1 and so
s = t.
Also, f is clearly onto K by the definition of K. So f is a bijection.
It remains to show that f is operation preserving.
Again let s and t be in H.
f(st) = a^-1 st a = (a^-1 s a) (a^-1 t a) = f(s)f(t)
QED
(b) immediately follows from the fact that H is a bijection.
Let s and t be in K. then s = a^-1 x a for some x in H, and t = a^-1 y a for some y in H.
so st = (a^-1 x a) ( a^-1 y a) = a^-1 xy a is in K since xy is in H.
And s^-1 = a^-1 x^-1 a is in K since x^-1 is in H.
So H is a subgroup.
A function f: G-->G where f(g) = a^-1 g a is known as an inner automorphism. The second part of (a) is to show that an inner automorphism takes a subgroup H to an isomorphic image of H = a^-1 H a.
Clearly f is 1-1 since if a^-1 s a = a^-1 t a then
a(a^-1 s a)a^-1 = a(a^-1 t a)a^-1 and so
s = t.
Also, f is clearly onto K by the definition of K. So f is a bijection.
It remains to show that f is operation preserving.
Again let s and t be in H.
f(st) = a^-1 st a = (a^-1 s a) (a^-1 t a) = f(s)f(t)
QED
(b) immediately follows from the fact that H is a bijection.