Nonabelian finite group G

In summary, the conversation discusses the possibility of finding two distinct elements, x and y, outside of a proper subgroup S in a nonabelian finite group G, such that the cosets Sx and Sy are equal to their inverses. This would mean that both x and y belong to a coset of S, while their inverses belong to a different, disjoint coset of S. The question is whether this is always possible in any nonabelian finite group. The conversation also suggests multiplying both cosets by x to get Sx^2=S and considers the implications for x^2.
  • #1
nbruneel
3
0
Hi,

This is not a homework question. I am a trying to prove a result for myself, and the question is can I always find, in a nonabelian finite group G, and some fixed proper subgroup S < G, two distinct elements, which we shall call x and y, outside of S, such that the cosets Sx = Sx^{-1}, and Sy = Sy^-1. That is, can we always find elements x, y outside of S such that x and its inverse x^{-1} both belong to some coset of S, while y and y^{-1} belong to a different, disjoint coset of S.

Nici.
 
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  • #2


If Sx=Sx-1 multiply both cosets by x to get Sx2=S. What does this tell us about x2?
 

1. What is a nonabelian finite group G?

A nonabelian finite group G is a mathematical structure that consists of a finite set of elements and a binary operation that combines any two elements in the set to form a third element. This operation does not need to be commutative, meaning that the order in which the elements are combined matters. Nonabelian groups are named after the mathematician Niels Henrik Abel, who studied commutative groups.

2. How is a nonabelian finite group G different from an abelian group?

The main difference between a nonabelian finite group G and an abelian group is that the binary operation in an abelian group is commutative, meaning that the order in which the elements are combined does not matter. In a nonabelian group, the order does matter, and the group may exhibit more complex behavior.

3. What are some examples of nonabelian finite groups?

Some examples of nonabelian finite groups include the dihedral groups, the symmetric groups, and the alternating groups. The dihedral groups are the symmetries of regular polygons, the symmetric groups are the permutations of a set, and the alternating groups are a subgroup of the symmetric groups.

4. How are nonabelian finite groups used in science?

Nonabelian finite groups have applications in many areas of science, including physics, chemistry, and computer science. In physics, they are used to describe the symmetry of physical systems, such as crystal structures. In chemistry, they are used to describe the arrangement of atoms in molecules. In computer science, they are used in cryptography and error-correcting codes.

5. What are some properties of nonabelian finite groups?

Nonabelian finite groups have many interesting properties, including closure, associativity, identity element, inverse element, and the existence of a finite order. They also have subgroups and cosets, and can be classified by their order and structure. Nonabelian finite groups are a rich area of study in mathematics and have many applications in various fields.

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