Correspondence Theorem for Normal Subgroups in Groups of Order 168

In summary, if G is a group of order 168 with a normal subgroup of order 4, then it also has a normal subgroup of order 28. The correspondence theorem states that there is a one-to-one correspondence between subgroups of G containing H and subgroups of G/H. Since G has a normal subgroup of order 4, G/H has a normal subgroup of order 7. Therefore, G has a normal subgroup of order 28.
  • #1
tyrannosaurus
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Homework Statement



Show that if G is a group of order 168 that has a normal subgroup of order 4, then G has a normal subgroup of order 28.


Homework Equations





The Attempt at a Solution


Let H be a normal subgroup of order 4. Then |G/H|=42=2*3*7, so then G?N has a unique, and therefore normal Sylow 7-subgroup, let's call it K.
I was told to use the correspondence theorem, but I am not sure where it works in here. any ideas?
 
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  • #2
Why don't you start by stating what the correspondence theorem says? The result you are seeking is an immediate consequence.
 

FAQ: Correspondence Theorem for Normal Subgroups in Groups of Order 168

1. What is a normal subgroup?

A normal subgroup is a subgroup that remains unchanged under conjugation by elements of its parent group. In other words, for any element in the parent group, if you apply it to a normal subgroup, the result will still be within the normal subgroup.

2. How is a normal subgroup different from a regular subgroup?

A normal subgroup is a specific type of subgroup that satisfies the condition of remaining unchanged under conjugation. Regular subgroups do not necessarily have this property.

3. Why are normal subgroups important in group theory?

Normal subgroups are important because they allow for the creation of quotient groups, which can help simplify the study of larger groups. They also have several applications in algebra and geometry.

4. What are some examples of normal subgroups?

The trivial subgroup and the parent group itself are always normal subgroups. Other examples include the center of a group, the commutator subgroup, and the kernel of a group homomorphism.

5. How are normal subgroups used in real-world applications?

Normal subgroups have applications in many areas, such as cryptography, physics, and chemistry. For example, they are used in coding theory and in the study of particle symmetries.

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