Normal subgroup of a product of simple groups

In summary, the conversation discusses a problem from Jacobson Algebra I regarding proving that every proper normal subgroup of a group G = G1 x G2, where G1 and G2 are simple groups, is isomorphic to either G1 or G2. Possible solutions involve considering the intersections of K with G1 x {1} and {1} x G2, and determining what K would be in each of the three cases: both intersections are trivial, both intersections are the full group, or only one intersection is trivial.
  • #1
john_nj
3
0

Homework Statement



This is an exercise from Jacobson Algebra I, which has me stumped.
Let G = G1 x G2 be a group, where G1 and G2 are simple groups.
Prove that every proper normal subgroup K of G is isomorphic to G1 or G2.

Homework Equations





The Attempt at a Solution


Certainly the intersection of K with G1 x {1} is normal, and so is isomorphic to the trivial group {1} or to G1. Similarly, the intersection of K with {1} x G2 is isomorphic to {1} or G2. But anyway this falls well short of a solution.

Thanks,

John

 
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  • #2
Well, you've basically got it. Try each of the three cases... if both intersections are trivial, then what is K? If both intersections are the full group, then what is K? And if only one intersection is trivial, what is K?
 
  • #3
Thank you. It's now clear.
 

What is a normal subgroup of a product of simple groups?

A normal subgroup of a product of simple groups is a subgroup that is invariant under conjugation by all elements of the product. In simpler terms, it is a subgroup that remains unchanged when acted upon by elements of the product group.

How is a normal subgroup of a product of simple groups different from a regular subgroup?

A normal subgroup of a product of simple groups is different from a regular subgroup because it is closed under conjugation by all elements of the product group, whereas a regular subgroup may not be. This means that a normal subgroup is a more restrictive form of a subgroup.

Why is the concept of normal subgroups important in group theory?

The concept of normal subgroups is important in group theory because it allows for the creation of quotient groups, which are crucial in studying the structure of groups. Normal subgroups also have important applications in other areas of mathematics, such as algebraic geometry and number theory.

What are some properties of normal subgroups of a product of simple groups?

Some properties of normal subgroups of a product of simple groups include the fact that the intersection of two normal subgroups is also a normal subgroup, and that the quotient group formed by a normal subgroup is isomorphic to a subgroup of the original group.

Can a product of simple groups have more than one normal subgroup?

Yes, a product of simple groups can have more than one normal subgroup. In fact, every group has at least two normal subgroups - the trivial subgroup and the whole group itself. The number of normal subgroups a group has can provide insight into its structure and properties.

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