Linked Questions

5
votes
4answers
566 views

Why in Quotient Group we need normal subgroup? [duplicate]

For a group $G$ and a normal subgroup $N$ of $G$, the quotient group of $N$ in $G$, written $G/N$ and read "$G$ modulo $N$", is the set of cosets of $N$ in $G$. Question : Why in the defintion of ...
0
votes
1answer
870 views

Proving that if left coset multiplication is well-defined then $H$ is a normal subgroup [duplicate]

I am given the left coset multiplication rule which is : $$aHbH=abH$$ Taking representatives $x \in aH $ and $a^{-1} \in a^{-1}H$ $$xHa^{-1}H=xa^{-1}H$$ Take representatives $a\in aH$ and $a^{-1}\...
1
vote
2answers
78 views

Why do normal subgroups “work”? [duplicate]

This is something I've wondered about for a while now; What is so special about normal subgroups that makes modding out by them "act nice"? I understand the proofs for things like the first ...
0
votes
0answers
54 views

Normal subgroup implies quotient group? [duplicate]

I was reading through a proof of the first isomorphism theorem on proofwiki and I didn't understand a line that said "by kernel is a normal subgroup of domain, $G_1/K$ exists", where $K = ker(\phi)$ ...
1
vote
0answers
49 views

Is G/H a group even if H is a subgroup of G but not a normal subgroup? [duplicate]

I know we can have left and right cosets without having the group be normal, but can we mod out by a subgroup without it being normal and still get a group? Thank you in advance!
0
votes
0answers
48 views

Cosets form a group if normal subgroup [duplicate]

Given a group G and its subgroup H that creates cosets, prove that cosets form a group iff H - is a normal subgroup. I've tried to find any good prove but I failed - most of the sources (including ...
20
votes
6answers
5k views

Intuition behind normal subgroups

I've studied quite a bit of group theory recently, but I'm still not able to grok why normal subgroups are so important, to the extent that theorems like $(G/H)/(K/H)\approx G/K$ don't hold unless $K$ ...
11
votes
5answers
1k views

Why doesn't this proof show that the operation on a factor group is well-defined?

Suppose $G$ is a group and let $H \triangleleft G$. Consider the factor group $G/H$ where the relation is $aHbH = abH$ for all $a,b \in G$. Suppose we wanted to show that the above relation is well-...
19
votes
3answers
2k views

Is there any intuitive understanding of normal subgroup?

As the define goes: A subgroup $N$ of a group $G$ is called a normal subgroup if it is invariant under conjugation; that is, for each element $n$ in $N$ and each $g$ in $G$, the element $gng^{−1}$ ...
42
votes
1answer
5k views

Quotient objects, their universal property and the isomorphism theorems

This is a question that has been bothering me for quite a while. Let me put between quotation marks the terms that are used informally. "Quotient objects" are always the same. Take groups, abelian ...
7
votes
3answers
11k views

Prime ideals and examples of them

So the question states that the intersection of two prime ideals is always a prime ideal. Well this is false but I need an example to counter it. I looked online and found one "For example, inside Z, ...
9
votes
4answers
623 views

Why is “working in $\mathbb {Z}_m$” essentially the same as “working with congruences modulo m”?

Due to my ignorance, I only superficially know the definition of congruence in number theory: For a given positive integer $n$, two integers $a$ and $b$ are called congruent modulo $n$, written $a\...
12
votes
3answers
16k views

Associativity, commutativity and distributivity of modulo arithmetic

Textbooks usually state "it is not hard to check that in modular arithmetic the usual associative, commutative and distributive properties continue to apply". Is there a way other than tedious proof ...
11
votes
2answers
2k views

Find all subrings of $\mathbb{Z}^2$

This may be a simple question: Find all subrings of $\mathbb{Z}^2$.
5
votes
3answers
257 views

Interpretation of ideals as unitless subsets

One way to prove that a field $K$ has no ideals except the entire field and the trivial ideal is to note the fact that every element $x$ has an inverse. By the definition of an ideal, if $x$ is in the ...

15 30 50 per page