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Compute in a group ring of symmetric group over Z/(3)


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.11

Consider the following elements of the group ring $\mathbb{Z}/(3)[S_3]$: $$\alpha = 1(2\ 3) + 2(1\ 2\ 3),\quad \beta = 2(2\ 3) + 2(1\ 3\ 2).$$ Compute $\alpha + \beta$, $2\alpha - 3\beta$, $\alpha\beta$, $\beta\alpha$, and $\alpha^2$.


Solution: Evidently, $$\alpha + \beta = 2(1\ 2\ 3) + 2(1\ 3\ 2)$$ $$2\alpha - 3\beta = 2\alpha = 2(2\ 3) + (1\ 2\ 3)$$ $$\alpha\beta = 2(1\ 2) + 1(1\ 2\ 3)$$ $$\beta\alpha = 1(1\ 3) + 2(1\ 3\ 2)$$ $$\alpha^2 = 1(1) + 2(1\ 2) + 2(1\ 3) + 1(1\ 3\ 2)$$


Linearity

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