I came across a GRE Mathematics Subject test question that said the following:
"Find the characteristic of the ring $Z_2 + Z_3$."
The explanation of the question starts with the statement that $$Z_2 + Z_3 = \{(0,0), (0,1), (0,2), (1,0), (1,1), (1,2)\}$$ My question is whether or not $Z_2 + Z_3$ is the proper way to label this set. Every other question I've come across using sets like this uses $Z_2 \times Z_3$. Is $Z_2 + Z_3$ the same thing as $Z_2 \times Z_3$ or is this an error in the text? Furthermore, if $Z_2 + Z_3$ is NOT the proper notation for this set, what would be the mathematical significance of adding these two groups (assuming $Z_n = \{0, 1, 2, \dots, n-1\}$ under addition modulo n)? Can groups be added, subtracted?
Also within the same text I came across a similar question regarding invariant subgroups: If H and K are both invariant subgroups of G (i.e. if $gH=Hg$ and $gK=Kg$ for every g in G), is HK also an invariant subgroup? Again, I'm not totally understanding the mathematical significance of HK. Does that mean $H \times K$ or something totally different?
I've found that the book I'm using has a few other errors that have been noted here and on other websites, so I want to be sure I'm thinking about this the proper way. Any answers, links, or recommended reading would be greatly appreciated!
If $H$ and $K$ are abelian groups (where the operation is usually written as addition), then it is standard to write $H\oplus K$ instead of $H\times K$. $H+K$ is... well, it's clear what's meant, but it's strange and non-standard.
If $H$ and $K$ are both subgroups of a third abelian group, then $H+K$ should describe the internal sum $\{h+k\mid h\in H, k\in K\}$, which is only isomorphic to $H\oplus K$ if $H\cap K = \{0\}$. So this is a practical reason to avoid the notation $H+K$ when referring to the direct sum/direct product of two abelian groups.
For general (possibly non-abelian) groups, we should never write $H+K$, because it's bad practice to think of nonabelian operations additively. For general groups, the notation $HK$ is the same thing, just written multiplicatively: $HK = \{hk \mid h\in H, k\in K\}$. This is standard, though extra conditions are required for it to actually be a group.