What do these group theory notations mean: $\overline{3}\otimes\overline{2}$, $\overline{2}\oplus\overline{3}$

Question

Can you explain or give a good reference to explain notations like $$\Large\overline{3}\otimes\overline{2}\qquad\qquad \overline{2}\oplus\overline{3}$$ and combinations of these. Thank you.

Answer 1

Any notation in mathematics is dependent on context, which you did not provide much of. (Where are you seeing these symbols - in your textbook? If so, which textbook? What is the surrounding sentence, or paragraph, where these notations show up?)

However, given that you've said this is group theory, what seems likely to me is that you are learning about the groups $\mathbb{Z}/n\mathbb{Z}$ (also known as $\mathbb{Z}_n$; in other words, modular arithmetic). Here is the relevant Wikipedia page.

If we're working in $\mathbb{Z}/n\mathbb{Z}$, then it's not uncommon to see these meanings:

• For any integer $a$, the notation $\overline{a}$ refers to the coset $$a+n\mathbb{Z}=\{\ldots,\;\;a-2n,\;\;a-n,\;\;a,\;\;a+n,\;\;a+2n,\;\;\ldots\}$$ which is an element of $\mathbb{Z}/n\mathbb{Z}$.

• The notation $\oplus$ refers to the addition of cosets, $$\overline{a}\oplus\overline{b}=\overline{a+b}$$

• The notation $\otimes$ refers to the multiplication of cosets, $$\overline{a}\otimes\overline{b}=\overline{ab}$$