# What does the plus sign contained in a circle ($\oplus$) mean in this case?

The fundamental group of the torus is isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$.

I know that the $\oplus$ symbol is the exclusive or symbol but I don’t understand how two of the same sets are XOR to each other.

Sorry if this is a very simple question.

#### Solutions Collecting From Web of "What does the plus sign contained in a circle ($\oplus$) mean in this case?"

The symbol $\oplus$ means direct sum.

The direct sum of two abelian groups $G$ and $H$ is the abelian group on the set $G\times H$ (cartesian product) with the group operation given by $(g,h) + (g’,h’) = (g+g’,h+h’)$.

You may well have seen this group denoted $G\times H$ and indeed, as long as the number of terms is finite, the direct sum and direct product of abelian groups are isomorphic.

More precisely, the direct sum is the coproduct in the category of abelian groups, while the direct product is the product.

$\oplus$ denotes the direct sum.

http://en.wikipedia.org/wiki/Direct_sum