Proving N(H) is a Subgroup of G to Normalizers in Group Theory

  • Thread starter wegmanstuna
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In summary, the normalizer N(H) of a subgroup H in a group G is defined as the set of elements a in G such that H^a = {x∈ G / axa^-1 ∈ H}. To show that N(H) is a subgroup of G, we need to prove that it satisfies the properties of closure, identity, and inverses. To show closure, we need to consider the elements a and b in N(H) and determine H^(ab). By writing down H^(ab), we can then show that H^{ab} is a subset of H and H is a subset of H^{ab}. This shows that N(H) is closed under the group operation and thus satisfies the property of closure
  • #1
wegmanstuna
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For a subgroup H of G and a fixed element a ∈ G,
let H^a = {x∈ G / axa^-1 ∈ H}, it's normalizer N(H) = {a∈G / H^a=H}

Show that for any subgroup H of G, N(H) is a subgroup of G.



I know that for the first one I need to show that closure holds, an identity exists, and inverses exist. But I don't even know where to start with closure!
Help!
 
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  • #2
If a and b are in N(H), write down what H^(ab) is. Now maybe you can try showing that [itex]H^{ab} \subset H[/itex] and [itex]H \subset H^{ab}[/itex].
 

FAQ: Proving N(H) is a Subgroup of G to Normalizers in Group Theory

What is a normalizer and why is it important?

A normalizer is a statistical method used to transform data into a standardized format. It is important because it allows for fairer comparisons between different datasets by removing any differences in scale or units.

What are some common types of normalizers?

Some common types of normalizers include min-max normalization, z-score normalization, and decimal scaling normalization. Other techniques such as log transformation and power transformation can also be used for normalization.

When should I use a normalizer?

A normalizer should be used when there are significant differences in scale or units between different datasets. This is necessary to ensure fair comparisons and accurate analysis of the data.

Are there any drawbacks to using a normalizer?

One potential drawback of using a normalizer is that it can sometimes distort the data or make it difficult to interpret. It is important to carefully select the appropriate type of normalizer and understand its effects on the data.

How do I choose which type of normalizer to use?

The type of normalizer to use depends on the specific characteristics of the data and the goals of the analysis. It is important to consider the distribution of the data, the presence of outliers, and the desired outcome when selecting a normalizer.

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