Normal subgroups - help me please

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In summary, a normal subgroup is a special type of subgroup of a group that is closed under conjugation. It is different from a regular subgroup in that it must be closed under conjugation, while a regular subgroup may not be. Normal subgroups have significant importance in group theory and abstract algebra, as they help us understand the structure and properties of groups and can be used to define important concepts. There are various methods for determining if a subgroup is normal, including checking if it is invariant under conjugation or using the subgroup criterion. A subgroup cannot be both normal and non-normal, as it is either one or the other. If a subgroup is non-normal, it means that there is at least one element in the original group that does not
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afirican
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If N is a normal subgroup of G and |G/N| = m, show that x^m is in N for all x in G
 
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Well, clearly G/N should have some pivotal role in this statement. Maybe you should move the problem from G to G/N to see if that helps.
 
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Sure, no problem! Let's start by defining some terms to make sure we're on the same page.

A subgroup N of a group G is called a normal subgroup if for every element x in G, xN = Nx. In other words, the left and right cosets of N are equal.

Now, let's look at the given information. We know that N is a normal subgroup of G, and |G/N| = m. This means that there are m distinct cosets of N in G.

Now, let's take any element x in G. Since N is a normal subgroup, we know that xN = Nx. This means that xN is one of the m distinct cosets of N in G.

Since there are m distinct cosets, we can write the product of all m cosets as (xN)^m. By definition, this product is equal to x^mN^m.

Now, since N is a subgroup, N^m is also a subgroup of G. And since N is a normal subgroup, we know that N^m is also a normal subgroup of G.

Therefore, x^mN^m is also equal to Nx^m. But we already know that xN = Nx, so this means that x^mN^m = Nx^m.

But we also know that Nx^m is one of the m distinct cosets of N in G. This means that x^mN^m is equal to one of the m distinct cosets of N in G.

But since N is a subgroup, this means that x^mN^m is equal to N itself. And since N is a normal subgroup, this means that x^m is in N for all x in G.

Hope this helps! Let me know if you have any other questions.
 

What is a normal subgroup?

A normal subgroup is a subgroup of a group that is invariant under conjugation by any element of the original group.

How is a normal subgroup different from a regular subgroup?

A normal subgroup is a special type of subgroup that is closed under conjugation, while a regular subgroup may not necessarily be closed under conjugation.

What is the significance of normal subgroups?

Normal subgroups play a crucial role in the study of group theory and abstract algebra. They help us understand the structure and properties of groups, and can also be used to define important concepts such as quotient groups and group homomorphisms.

How can I determine if a subgroup is normal?

There are several methods for determining if a subgroup is normal. One approach is to check if the subgroup is invariant under conjugation by all elements of the original group. Another method is to use the subgroup criterion, which states that a subgroup is normal if and only if it contains all of the conjugates of its elements.

Can a subgroup be both normal and non-normal?

No, a subgroup cannot be both normal and non-normal. A subgroup is either normal or non-normal, there is no in-between. If a subgroup is not normal, it means that there exists at least one element in the original group that does not preserve the subgroup under conjugation.

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