Quotient Groups & how to interpret notation?

[mathjam0990]

Hello,

I am having some trouble truly interpreting what certain notation means when defining quotient groups, etc. (My deepest apologies in advance, with my college workload I simply have not had the time to really sit down and master latex.) Here are a few random examples I've seen in textbooks, online, etc.

example 1) What does Z/_6z (z mod 6z) really mean? Is this like the group of integers such that every integer is divisible by 6?

example 2) What is R^x ? Is this the group of real numbers under multiplication? Do we include 0 and 1?

example 3) If we simply just write R, it that the group of real numbers under both multiplication and addition?

example 4) Given D₄ is a dihedral group, why is the group D₄/H = [H, rH} isomorphic to Z/_2z ? Is this because Z/_2z is the group of only even integers and the number of elements in D₄/H equals 2 which is even, thus they are isomorphic?

These are just a few of several examples. Obviously I cannot go on forever, but I guess I am just hoping to get a better understanding of what all this notation means.

Thank you in advance!

 

[Deveno]

Hello,

I am having some trouble truly interpreting what certain notation means when defining quotient groups, etc. (My deepest apologies in advance, with my college workload I simply have not had the time to really sit down and master latex.) Here are a few random examples I've seen in textbooks, online, etc.

example 1) What does Z/_6z (z mod 6z) really mean? Is this like the group of integers such that every integer is divisible by 6?

This is more properly written $\Bbb Z/6\Bbb Z$ and represents the quotient of the additive group $(\Bbb Z,+)$ by its (normal) subgroup $6\Bbb Z$, the subgroup of all multiples of $6$. Basically, we "identify" all multiples of $6$ calling them "equivalent to $0$", so that we wind up with $6$ cosets:

$0 + 6\Bbb Z = 6\Bbb Z = \{\dots,-12,-6,0,6,12,18,\dots\}$
$1 + 6\Bbb Z = \{\dots,-11,-5,1,7,13,19,\dots\}$
$2 + 6\Bbb Z = \{\dots,-10,-4,2,8,14,20,\dots\}$
$3 + 6\Bbb Z = \{\dots,-9,-3,3,9,15,21,\dots\}$
$4 + 6\Bbb Z = \{\dots,-8,-2,4,10,16,22,\dots\}$
$5 + 6\Bbb Z = \{\dots,-7,-1,5,11,17,23,\dots\}$

We add two cosets $(k + 6\Bbb Z) + (m + 6\Bbb Z)$ by finding the coset $k+m$ is in, so for example:

$(3 + 6\Bbb Z) + (5 + 6\Bbb Z) = 2 + 6\Bbb Z$, since $8 \in 2 + 6\Bbb Z$.

It's hard to be "clearer" without going into great detail of what a quotient group actually is. But I'll try to answer more questions about them, if you ask.

example 2) What is R^x ? Is this the group of real numbers under multiplication? Do we include 0 and 1?

Given a ring $R$, the notation $R^{\times}$ is typically used for the multiplicative group of units of $R$. For example, in the ring of integers, the group of multiplicative units is $\{-1,1\}$. If $R = \Bbb R$, the real numbers, then the multiplicative group of units is the non-zero real numbers, since given $r \neq 0 \in \Bbb R$, we have an $s \in \Bbb R$ such that:

$rs = sr = 1$, namely $s = \dfrac{1}{r}$.

example 3) If we simply just write R, it that the group of real numbers under both multiplication and addition?

$\Bbb R$ is not a group under multiplication, since $0$ has no (multiplicative) inverse. So if someone refers to "the group $\Bbb R$", they usually mean under the operation of addition.

example 4) Given D₄ is a dihedral group, why is the group D₄/H = [H, rH} isomorphic to Z/_2z ? Is this because Z/_2z is the group of only even integers and the number of elements in D₄/H equals 2 which is even, thus they are isomorphic?

$D_4$ could mean one of two groups (both dihedral) one has $4$ elements, and one has $8$ elements. To be sure which one you have here, I'd need to know what "$H$" is. In any case, it turns out that ANY two groups with two elements are isomorphic. $\Bbb Z/2\Bbb Z$ is *not* the group of only even integers, that is $2\Bbb Z$. $\Bbb Z/2\Bbb Z$ has two cosets (two elements):

$0 + 2\Bbb Z = 2\Bbb Z$ -"even integers"
$1 + 2\Bbb Z$ -"odd integers".

We add these cosets using the usual "rules":

odd + odd = even
odd + even = even + odd = odd
even + even = even

You may want to convince yourself that this is same rule as I exhibited with $\Bbb Z/6\Bbb Z$ above with $2$ in place of $6$.

These are just a few of several examples. Obviously I cannot go on forever, but I guess I am just hoping to get a better understanding of what all this notation means.

Thank you in advance!

It sounds like you don't have a good grasp of how a quotient group is fundamentally different than a subgroup. You may have to back up a few steps, and see where you lost the thread.

 

[mathjam0990]

Thank you a million times for your answer. It makes much more sense than before. I only know by definition a quotient group is left cosets so G/H = {gH for all g in G} I think. And a subgroup must have closure, identity and inverse. But, when it comes to applying these definitions that is where I kind of get lost. I have some more reading up on this to do, but this answer gave me a great start. Thank you.