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envision October 3rd, 2009 06:35 PM

Subgroup/Normal Subgroup/Automorphism Questions
 
In this problem, assume (important!) that G is an Abelian. Set H = {g in G : g^5 = e}. (Warning: Expressions such as x^1/5 are not well defined. Do not use them!)

(a) Show H is a subgroup of G.
(b) Show H is a normal subgroup of G.
(c) Assume further that G is finite and that H = {e}. Show that
the map phi : G --> G given by phi(g) = g^5 is an automorphism.

(Definition: An automorphism is an isomorphism from a group to itself.)

cknapp October 4th, 2009 03:12 AM

Re: Subgroup/Normal Subgroup/Automorphism Questions
 
Quote:

Originally Posted by envision
In this problem, assume (important!) that G is an Abelian. Set H = {g in G : g^5 = e}. (Warning: Expressions such as x^1/5 are not well defined. Do not use them!)

(a) Show H is a subgroup of G.

Use a subgroup test.

Quote:

(b) Show H is a normal subgroup of G.
You need to show that Hg = gH for all g in G. This comes almost directly from the Abelian-ness of G.
Quote:

(c) Assume further that G is finite and that H = {e}. Show that
the map phi : G --> G given by phi(g) = g^5 is an automorphism.
Show that it is operation preserving. This gives you a homomorphism. Have you learned any properties of homomorphism yet? If so, there's one about the kernel of phi. This shows it's a bijection.
If not, show it is injective: show that a?b -> phi(a)?phi(b). An injective function from A->A must be bijective for any finite set A.


Cheers.

envision October 4th, 2009 08:14 PM

Re: Subgroup/Normal Subgroup/Automorphism Questions
 
Quote:

Originally Posted by cknapp
Quote:

Originally Posted by envision
In this problem, assume (important!) that G is an Abelian. Set H = {g in G : g^5 = e}. (Warning: Expressions such as x^1/5 are not well defined. Do not use them!)

(a) Show H is a subgroup of G.

Use a subgroup test.

Quote:

(b) Show H is a normal subgroup of G.
You need to show that Hg = gH for all g in G. This comes almost directly from the Abelian-ness of G.
[quote:16v4wr01]
(c) Assume further that G is finite and that H = {e}. Show that
the map phi : G --> G given by phi(g) = g^5 is an automorphism.

Show that it is operation preserving. This gives you a homomorphism. Have you learned any properties of homomorphism yet? If so, there's one about the kernel of phi. This shows it's a bijection.
If not, show it is injective: show that a?b -> phi(a)?phi(b). An injective function from A->A must be bijective for any finite set A.


Cheers.[/quote:16v4wr01]

concerning part c...
we've learned about homomorphisms. isn't the kernel of phi = e or something like that? i don't really know how to even begin showing part c.

cknapp October 4th, 2009 10:37 PM

Re: Subgroup/Normal Subgroup/Automorphism Questions
 
The kernel is the set of elements which map to e.
You should have learned that a homomorphism is n-1: every element in the range has the same number (n) of elements map to it.
If you can show that only one element maps to e, then you have a 1-1, homomorphism. And since the sets are finite and the same size, this is a bijective homomorphism-- an isomorphism.


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