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I am trying to self-study abstract algebra but I am having some difficulty with quotient groups. I understand cosets and the importance of using normal subgroups as they serve as the kernel of a homomorphism. What I fail to grasp intuitively is when books discuss "losing information" in the construction of a quotient group. I know that the subgroup you mod out by serves as the identity in the newly-constructed quotient group but how exactly are we losing information? I'm sure this is a very basic question and I'm overlooking something very simple but I would appreciate your help in understanding this.

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  • $\begingroup$ Notice that $\mathbb{Z}/2\mathbb{Z}$ has only two elements, but $\mathbb{Z}$ has infinitely many. $\endgroup$ – Daniel Aug 20 '16 at 14:38
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You lose information about the individual elements of the original group when you look only at their coset in the quotient group. $\Bbb Z /2\Bbb Z$ is a good example - if you know an integer's coset in this quotient group, you only know if the number is even or odd. That is still information, but not as much as if you knew the number itself. (For example, you wouldn't know from the coset if the number was divisible by 5).

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However, with information on quotient groups one can actually gain information about the original group. An example, suppose you know that $G/Z(G)$ is a $p$-group, then this yields that $G'$ is a $p$-group, even that $G$ is nilpotent. And there are many more of these examples. In fact, classes of groups are defined with certain series of (sub)quotients, like nilpotent, supersolvable, solvable groups. You will be amazed how many properties of the main group from these definitions can be derived!

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Example: Let $G= \Bbb{Z}/ 24 \Bbb{Z}$. Let's say that you measure the time with this group, so for example $14 + 24 \Bbb{Z}$ denotes the hour $14:00$, which is also $2 \mathrm{pm}$.

Now, you can quotient by $12 \Bbb{Z}/24 \Bbb{Z}$, and get the new group $$G/(12\Bbb{Z}/24 \Bbb{Z}) \cong \Bbb{Z}/12 \Bbb{Z}$$ this quotient tells you the time in cycles of $12$ hours. So, for example the element $2+ 12 \Bbb{Z}$ denotes $2 \mathrm{am}$ OR $2 \mathrm{pm}$. So, if I tell you that it's $2+ 12 \Bbb{Z}$: what time is it? Is it $2 \mathrm{am}$ or is it $2 \mathrm{pm}$? Well, you lost some information with the quotient, and you can't recover it. You would have had more information if I told you that it's $14 + 24 \Bbb{Z}$.

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  • $\begingroup$ Thank you all so much! All of your answers were very insightful. It makes sense now. $\endgroup$ – Steven Scott Aug 20 '16 at 15:09

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