Let G = D12, given by generators a,b with |a|=6, |b|=2, and ba=a-1b.

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In summary, the normalizer of the subgroup H in group G is either H or G. By Lagrange's theorem, the order of G is equal to the order of H times the index of H in G. Since the normalizer must contain H, its order must be greater than or equal to that of H. Therefore, the order of NG(H) cannot be 6, as 4 does not divide 6.
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Bachelier
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Let G = D12, given by generators a,b with |a|=6, |b|=2, and ba=a-1b. Let H = { 1, a3, b, a3b }. Find the normalizer of H in G and find the subgroups of G that are conjugate to H.

The normalizer of H is a subgroup containing H, so since H has index 3, either NG (H) = H or NG (H) = G.

I'm confused about this part.

By Lagrange, |G|=|H|[G:H], since normalizer must contain H, then NG(H)≥H. But then its order can be 6. I'm missing something here about [G:NG(H)]
 
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I honestly don't understand your question :frown:
Everything you say is true. So what's confusing you?
 
  • #3


no the order cannot be 6. NG(H) contains H as a subgroup. H is of order 4, and 4 does not divide 6.
 

1. What is the order of G?

The order of G is 12, since it is given by generators a and b with |a|=6 and |b|=2, which means that the smallest positive integer n such that a^n = e (the identity element) is 6, and the smallest positive integer m such that b^m = e is 2. The order of G is therefore the least common multiple of 6 and 2, which is 12.

2. What is the structure of G?

The structure of G is a dihedral group of order 12, denoted as D12. This means that it is a group of symmetries of a regular polygon with 12 sides, and it contains both rotations and reflections.

3. What are the elements of G?

The elements of G are: the identity element e, the elements a, a^2, a^3, a^4, a^5, b, ba, ba^2, ba^3, ba^4, and ba^5. These elements can be combined using the group operation, which in this case is composition of symmetries (i.e. rotations and reflections).

4. How do you represent the group operation in G?

The group operation in G is composition of symmetries. This means that to perform the group operation on two elements, we first apply one symmetry (rotation or reflection) and then apply the other. For example, to perform the operation a*b, we would first rotate by the amount specified by b, and then reflect according to a. In this case, since |a|=6 and |b|=2, the rotation specified by b would be 180 degrees (or a half-turn), and the reflection specified by a would be a rotation of 60 degrees followed by a reflection.

5. How do you determine if a given element is in G?

To determine if a given element is in G, we can use the generators a and b to generate all the elements of G. If the given element can be expressed as a combination of these generators and their inverses (e.g. a^-1 = a^5), then it is in G. If not, then it is not in G. Alternatively, we can also check if the given element satisfies the group axioms of closure, associativity, identity, and inverse.

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