Abstract Algebra: Proving Normal Subgroup and Isomorphisms

In summary, the conversation discusses the definition of a group G and how it relates to two other groups, G1 and G2. The conversation also explores three parts of a homework problem, including showing that a subgroup N is normal in G, proving that N is isomorphic to G1, and showing that G/N is isomorphic to G2. The concept of isomorphism, which involves finding a one-to-one mapping between elements of two groups, is important in solving these problems.
  • #1
lorena82186
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0

Homework Statement


If G1, G2 are two groups and G = G1 times G2 = {(a,b) such that a is an element of G1, b is and element of G2}, where we define (a,b)(c,d) = (ac, bd),

(a) Show that N = {(a, e2) such that a is an element of G1}, where e2 is the unit element of G2, is a normal subgroup of G.

(b) Show that N is isomorphic to G1.

(c) Show that G/N is isomorphic to G2.


Homework Equations





The Attempt at a Solution


I did part (a) but I do not know how to begin parts (b) and (c)
 
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  • #2
understand what is the definition for isomorphism (ie. need to find a 1-1 mapping from elements in N to elements in G1 such that the multiplication table is the same)
 
  • #3
(b) write down the only conceivable map, and show it is an isomorphism

(c) see (b).
 

1. What is a normal subgroup?

A normal subgroup is a subgroup of a group that is invariant under conjugation by elements of the larger group. This means that if an element of the larger group is conjugated with an element of the normal subgroup, the result will still be an element of the normal subgroup.

2. How do you prove that a subgroup is normal?

To prove that a subgroup is normal, you must show that for every element in the larger group and every element in the subgroup, the conjugate of the latter by the former is also in the subgroup. This can be done by showing that the subgroup is closed under conjugation, or by using the definition of normal subgroup.

3. What is an isomorphism in abstract algebra?

An isomorphism is a bijective homomorphism between two algebraic structures. In abstract algebra, this means that two groups are isomorphic if there exists a one-to-one correspondence between their elements that preserves the group structure.

4. How do you prove that two groups are isomorphic?

To prove that two groups are isomorphic, you must show that there exists a bijective homomorphism between the two groups. This can be done by explicitly constructing the isomorphism or by showing that the groups have the same structure and properties.

5. Can normal subgroups and isomorphisms be combined in a proof?

Yes, normal subgroups and isomorphisms can often be combined in a proof. For example, if two groups are isomorphic and one has a normal subgroup, then the image of that subgroup under the isomorphism will also be a normal subgroup in the other group. This can be useful in proving properties of normal subgroups and isomorphisms simultaneously.

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