Prove that a group of order 42 has a nontrivial normal subgroup

In summary, to prove that a group of order 42 has a nontrivial normal subgroup, we can use Cauchy's Theorem to find an element of order 7. Letting A = <a>, where o(a) = 7, we can show that A is a normal subgroup by considering its conjugate subgroup B. If A ∩ B = 1, this would result in a contradiction, as it would imply ord(AB) = 49 > ord(G). Therefore, A ∩ B must equal 7, meaning that A = B, and thus A is normal.
  • #1
Viking85
2
0

Homework Statement



Prove that a group of order 42 has a nontrivial normal subgroup

Homework Equations



We are supposed to use Cauchys Theorem to solve the problem
We are not allowed to use any of Sylows Theorems


The Attempt at a Solution



By using Cauchys Theorem i know there there is an element of order 7
I then let A = <a> where o(a)=7
Essentially i know the normal subgroup is of order 7, but i need to still show it is normal
 
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  • #2
If A isn't normal then it has a conjugate subgroup B which also has seven elements. What's the order of AB?
 
  • #3
Well, I think that A ∩ B = 1 or 7 (Using Lagranges Theorem)
If A ∩ B is 1 it results in a contradiction because that would imply ord(AB)=49 > ord(G)

Which means that A ∩ B = 7

Which means the A = B ??
 
  • #4
Viking85 said:
Well, I think that A ∩ B = 1 or 7 (Using Lagranges Theorem)
If A ∩ B is 1 it results in a contradiction because that would imply ord(AB)=49 > ord(G)

Which means that A ∩ B = 7

Which means the A = B ??

Right. That makes A normal, yes?
 

What is the definition of a group?

A group is a mathematical structure consisting of a set of elements and a binary operation that combines any two elements to produce a third element. It must also satisfy four properties: closure, associativity, identity, and inverse.

What is the order of a group?

The order of a group is the number of elements in the group. It is denoted by |G|, where G is the group.

What is a nontrivial normal subgroup?

A nontrivial normal subgroup of a group is a subgroup that is both proper (not equal to the whole group) and normal (invariant under conjugation by any element of the group).

Why is it important to prove that a group of order 42 has a nontrivial normal subgroup?

Proving the existence of a nontrivial normal subgroup in a group of order 42 has important implications in group theory and other areas of mathematics. It can also provide insights into the structure of the group and its subgroups.

How do you prove that a group of order 42 has a nontrivial normal subgroup?

There are several methods for proving the existence of a nontrivial normal subgroup in a group of order 42. One approach is to use the Sylow theorems, which state that if a group has a prime power order, then it must have subgroups of that prime power order. Another approach is to use group actions and the orbit-stabilizer theorem to show that there must be a nontrivial normal subgroup.

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