Proving Cosets of Subgroups in Nonabelian Finite Groups

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In summary, Nici is trying to prove a result in a nonabelian finite group where there is a fixed proper subgroup S and two distinct elements x and y outside of S such that Sx = Sx^{-1} and Sy = Sy^{-1}. The question is whether this is always possible to find in any nonabelian finite group, specifically in Z/6 with S = {0,2,4}.
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nbruneel
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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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Consider Z/6 with S= {0,2,4}. S1(=S5) & S3 are not disjoint.
 

Related to Proving Cosets of Subgroups in Nonabelian Finite Groups

1. What are cosets of subgroups in nonabelian finite groups?

Cosets of subgroups in nonabelian finite groups are a fundamental concept in group theory. A coset is a subset of a group that is obtained by multiplying all elements of a subgroup by a fixed element of the group. These cosets help to partition the group into distinct subsets and allow for the study of the group's structure.

2. How do you prove the existence of cosets in nonabelian finite groups?

The existence of cosets in nonabelian finite groups can be proven using the Lagrange's theorem, which states that the order of a subgroup must divide the order of the parent group. This theorem allows us to show that the cosets obtained by multiplying a subgroup by a fixed element are all distinct and form a partition of the group.

3. What is the significance of proving cosets in nonabelian finite groups?

Proving cosets in nonabelian finite groups is essential in understanding the structure of these groups. It allows us to identify the number of distinct subgroups and their orders, which can provide insights into the group's properties and its relationship with other groups.

4. How do you show that two cosets in a nonabelian finite group are equal?

To show that two cosets in a nonabelian finite group are equal, we need to prove that they contain the same elements. This can be done by showing that every element in one coset can be obtained by multiplying an element in the other coset by a fixed element of the group. If this condition is satisfied, then the two cosets are equal.

5. Can cosets of subgroups in nonabelian finite groups be used to prove the group's commutativity?

No, cosets of subgroups in nonabelian finite groups cannot be used to prove the group's commutativity. This is because nonabelian finite groups, by definition, do not have the commutative property. However, cosets can be used to prove other important properties of groups, such as the existence of normal subgroups.

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