How do "Identify" Quotient Groups with Another Standard Group (Algebra Self Study)
Monday 23 March 2026 at 09:28
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Edited by Zafar Bakhromov, Wednesday 1 April 2026 at 13:44
A Quotient Group is the set of cosets of in , with a law of composition that makes it into a group. The whole idea is that, if we write the law of composition on as "", then we want to view two elements and that "differ by a multiple of " as the same element in .
While self-studying the book "Algebra 2.e." by Michael Artin, I found it somewhat difficult to grasp how to analyze the structure of a quotient group. Here I will summarize what I've learned through several exercises.
Exercise:
Let be the group of upper triangular real matrices , with and different from zero. For each of the following subsets, determine whether or not is a subgroup and whether or not is a normal subgroup. If is a normal subgroup, identify the quotient group .
(i) is the subset defined by .
(ii) is the subset defined by .
(iii) is the subset defined by .
Source: "Algebra" 2.e., Michael Artin, p.75
My Solution:
(i) is the set of diagonal matrices, with nonzero entries and on the diagonal. This forms a normal subgroup of , but we omit the proof to keep our focus on quotient groups.
Since , an element of looks like:
and we need to choose a representative element for each to analyze . With the entries and of free to choose for any , we may pick and to make multiplication easier. Hence, denoting a representative element of as , we get:
where we have defined . Thus, we have reduced into workable form.
Now, to analyze , we have to multiply two of its elements. Taking , we have that:
Hence, all multiplication does in is that it adds the entries in the top right corners of and , so it really looks like the addition of real numbers.
We are now ready to identify the quotient group with a standard group. Let , where denotes the additive group of real numbers. Then, is an isomorphism. The bijectivity of is given by the fact that can be any real number (including 0), and is a homomorphism because:
Hence, is isomorphic to .
(ii) We similarly have that is a normal subgroup of . For some particular , an element of looks like:
With the entries of free to choose, we may take and , in which case we get that:
so
Thus, by similar logic as before, , where is the multiplicative group of nonzero real numbers.
(iii) I'm getting a little tired of typesetting this so we're just going to note that by similar reasoning as in (ii).
Conclusion:
It took me about a week of working through examples like these to really grasp what a quotient group is doing. The key insight, for me, was not just the definition, but learning how to choose useful representatives of cosets. By simplifying each coset to a canonical form, the group operation in becomes explicit and often reveals a familiar structure. In this way, quotient groups stop feeling like abstract sets of cosets and instead become concrete objects that capture exactly what remains of a group after “modding out” a chosen symmetry.
How do "Identify" Quotient Groups with Another Standard Group (Algebra Self Study)
A Quotient Group is the set of cosets of in , with a law of composition that makes it into a group. The whole idea is that, if we write the law of composition on as "", then we want to view two elements and that "differ by a multiple of " as the same element in .
While self-studying the book "Algebra 2.e." by Michael Artin, I found it somewhat difficult to grasp how to analyze the structure of a quotient group. Here I will summarize what I've learned through several exercises.
Exercise:
Source: "Algebra" 2.e., Michael Artin, p.75
My Solution:
(i) is the set of diagonal matrices, with nonzero entries and on the diagonal. This forms a normal subgroup of , but we omit the proof to keep our focus on quotient groups.
Since , an element of looks like:
and we need to choose a representative element for each to analyze . With the entries and of free to choose for any , we may pick and to make multiplication easier. Hence, denoting a representative element of as , we get:
where we have defined . Thus, we have reduced into workable form.
Now, to analyze , we have to multiply two of its elements. Taking , we have that:
Hence, all multiplication does in is that it adds the entries in the top right corners of and , so it really looks like the addition of real numbers.
We are now ready to identify the quotient group with a standard group. Let , where denotes the additive group of real numbers. Then, is an isomorphism. The bijectivity of is given by the fact that can be any real number (including 0), and is a homomorphism because:
Hence, is isomorphic to .
(ii) We similarly have that is a normal subgroup of . For some particular , an element of looks like:
With the entries of free to choose, we may take and , in which case we get that:
so
Thus, by similar logic as before, , where is the multiplicative group of nonzero real numbers.
(iii) I'm getting a little tired of typesetting this so we're just going to note that by similar reasoning as in (ii).
Conclusion:
It took me about a week of working through examples like these to really grasp what a quotient group is doing. The key insight, for me, was not just the definition, but learning how to choose useful representatives of cosets. By simplifying each coset to a canonical form, the group operation in becomes explicit and often reveals a familiar structure. In this way, quotient groups stop feeling like abstract sets of cosets and instead become concrete objects that capture exactly what remains of a group after “modding out” a chosen symmetry.