Question

Group Theory Question. Can a right coset strictly contain a left coset?

More precisely, is there a group \(G\), with subgroup \(H\), and an element \(g \in G\), such that \(gH \subseteq Hg\) but \(gH \neq Hg\)? I understand the answer is clearly "no" if G is finite, since |gH| = |Hg| always, so then \(gH \subseteq Hg\) clearly implies \(gH = Hg\). But for infinite groups this is not enough...

Thanks!

Answer 1

@zzr0ck3r @KingGeorge The only people on here who I know post group theory questions... :)

Answer 2

would have to come with a non abelian infinite group where \(gH\subset Hg\)

Answer 3

but i can't think of one off the top of my head where containment is strict

Answer 4

matrices come to mind

Answer 5

Indeed. And stay away from normal subgroups altogether...

I thought perhaps \(GL_n(\mathbb{R})\) might be a good place to start looking. Not sure how to start looking though!

I was rather hoping someone might know some facts which would prove that the answer was no.

Answer 6

i think we can come up with an example, may take a minute ( or more )

Answer 7

\[H=\left[ {\begin{array}{cc}
a & 1 \\
0 & 1 \\
\end{array} } \right]\] maybe? and \(g=\left[ {\begin{array}{cc}
2 & 0 \\
0 & 1 \\
\end{array} } \right]\)

Answer 8

maybe not
but i actually remember something like this with linear maps
\(G\) is the set of linear maps \(g(x)=ax+b\)

Answer 9

yeah the gimmick is to make \(H\) be the subgroup where \(b\) is an integer
hold on a second

Answer 10

Linear maps? Is the group operation composition?

Answer 11

right

Answer 12

i forget exactly how it works though

Answer 13

some how you get something is an integer and something isn't so containment is strict

Answer 14

i know \(H\) is the set where \(h(x)=ax+n, n\in m\mathbb{Z}\)

Answer 15

the m was a typo, \(H\) is the set of elements \(h(x)=ax+n,n\in \mathbb{Z}\)

Answer 16

That makes sense actually. I was hoping to get something with integers and something with only even integers or something.

Answer 17

ok lets take \(g\) as \(g(x)=2x+3\) maybe that will work

Answer 18

lol thank god for google
here is a much better explanation
see example 2 here
http://math.stackexchange.com/questions/460247/example-of-a-group-with-non-equal-coset-for-a-fixed-element

Answer 19

Wow. You put my googling abilities to shame...

Answer 20

wish they had had this when i was in school
would have 2 or 3 PhD by now

Answer 21

Ah, but where's the fun when you can just google the answer instead of working things out? ;-)

Answer 22

well we have the satisfaction at least of knowing we were on the right track
examples like this are usually canonical examples, the one or two you know but probably would not find on your own without significant hints

Answer 23

how many continuous nowhere differentiable functions do you know?
or uncountable sets of measure zero?

Answer 24

I suppose so. I wonder how people came up with these. Experimentation? Experience? I got as far as looking for a non-abelian infinite group. Everything after that seemed to be "try some well-known groups and hope for the best"!

Answer 25

my guess is either none or one

Answer 26

you get the book called "counter examples in analysis" or "counter examples in algebra"

Answer 27

I've heard of the Wierstrass function (or however its spelt). That's all I know. As for the second question, I don't know what "measure zero" means!

Answer 28

find the thing that "is this but not that", proving that the two sets are not equivalent

Answer 29

you will know it soon enough i am sure
means has no length basically
or its probability is zero
the example you will know will be called the cantor set, and the only other ones you will know will be variations on that theme
have fun

Answer 30

Sounds interesting! I look forward to (second year) maths next term...

Is there a book "Counterexamples in Algebra"? I've seen a "Counterexamples in Analysis" book, but not for algebra.