Question about Cosets and Lagrange's Theorem

In summary, the conversation discusses whether Ha, a right coset of a subgroup H generated by element a, is also a subgroup. The textbook notation may suggest that Ha is a subgroup, but the speaker argues that it may not be closed under the operation unless a is an element of H. However, it is later clarified that Ha is a subgroup if and only if it is equal to H and a is an element of H. The disjoint union of H and other cosets in G is also mentioned, with the reminder that only H is a subgroup.
  • #1
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If a is an element of G and H is a subgroup of H, let Ha be the right coset of H generated by a.

is Ha a subgroup?

I have this question because i feel like the answer should be know, yet my textbook notation makes it look like it is.

Why I think it should only be a set:
Let G = <a> and H = <a^3>. and |a| = 6,.
Then H = {1,a^3}, Ha = {a,a^4} Ha^2 = {a^2,a^5}.
To me it looks like Ha and Ha^2 can not be subgroups because they are not closed under the operation, unless the operation is multiplying the elements by a^3.. I don't know I'm confused.

But then I think they should be considered subgroups, because it goes on to say that Ha = H iff a is in H. So Ha must be a subgroup if it equals a subgroup... Anyway, anyone have insight here?
 
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  • #2
The problem doesn't not say that [itex] a [/itex] must be an element of [itex] H [/itex]. What the textbook goes on to say about the case when [itex] a [/itex] is an element of [itex] H [/itex] is not relevant to the problem.
 
  • #3
PsychonautQQ said:
is Ha a subgroup?
##Ha## is a subgroup if and only if ##Ha = H## if and only if ##a \in H##.

One easy way to see this is that if ##Ha \neq H## then ##Ha## and ##H## are disjoint, so ##Ha## does not even contain the identity element.
 
  • #4
so when you create G out of disjoint cosets, you are joining H (a subgroup of G) with Ha, Hg, Hh, etc etc, which are only disjoint sets of elements of G?
 
  • #5
PsychonautQQ said:
so when you create G out of disjoint cosets, you are joining H (a subgroup of G) with Ha, Hg, Hh, etc etc, which are only disjoint sets of elements of G?
Yes, ##G## is the disjoint union of ##H## and the other cosets of ##H##. Only ##H## is a subgroup. The other cosets are subsets of ##G## with the same size as ##H##, but they aren't subgroups.
 
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What is a coset?

A coset is a subset of a group that is formed by multiplying each element of the subgroup by a fixed element of the group. It is denoted by a left or right multiplication of the subgroup by the fixed element.

How is Lagrange's Theorem related to cosets?

Lagrange's Theorem states that the order of a subgroup must divide the order of the group it is contained in. In relation to cosets, this means that the number of elements in a coset must divide the number of elements in the subgroup it is formed from.

Can a coset contain more than one subgroup?

No, a coset can only contain one subgroup. However, a subgroup can contain multiple cosets.

How can cosets be used in group theory?

Cosets are useful in group theory because they allow us to partition a group into distinct subgroups. This can help us understand the structure of a group and prove theorems, such as Lagrange's Theorem.

What is the difference between a left coset and a right coset?

The only difference between a left coset and a right coset is the order in which the elements are multiplied. In a left coset, the fixed element is multiplied on the left side of the subgroup, while in a right coset, it is multiplied on the right side. However, the resulting cosets will contain the same elements.

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