Symmetry group 3 and writing elements in terms of each other

Question

In the group $S_3$ and let $a,b$ be distinct elements of order two. Express each non-identity element in $S_3$ as a word in $a,b$.

I am simply confused by what this means. I know $S_3 = \{e, (1\; 2), (1\; 3), (2\; 3), (1\; 2\; 3), (1 \;3 \;2)\}$, but I am struggling with how to express the other terms as "words."

Answer 1

By word, they just mean a string (ie a product) of $a$ and $b$. As $a$ and $b$ are the elements of order 2, you can just arbitrarily say that $a$ is (12) and $b$ is (23). Now write the other elements as products of $a$ and $b$. For example what element is $ab$?

Answer 2

So, as it turns out, any two of $(12),(13)$ and $(23)$ generate $S_3$.

That just means that any element of $S_3$ can be written as a finite product of, say $(12)$ and $(13)$ 's. (We should consider the inverses of these elements as well. But of course they are their own inverses.)

So let's see: $e=(12)(12),(12)=(12),(13)=(13),(23)=(12)(13)(12),(123)=(13)(12)$.

And I leave $(132)$ for you.