First, a little motivation:
I have read the section on Group Actions in Dummit & Foote, the wikipedia page, and (countably many) other references. And seemingly without exception, they only offer rote and/or abstract examples, such as:
- Let $ga = a$ for al $ g \in G, a \in A$
- The symmetric group $S_N$ acting on $A$ by $\sigma \cdot a = \sigma(a)$
- Something about regular n-gons and $D_{2n}$
- $g \cdot a = ga$...
I don't mean to undermine the importance of these examples, but I'm left with no hands-on experience with these things. Exercise $\S$ 1.7.8(b) in D&F says:
"Describe explicitly how the elements $(1 \ 2)$ and $(1 \ 2 \ 3)$ act on the six 2-element subsets of $ \left \{1, 2, 3, 4 \right \}$.
How does a three cycle permute two elements?
Furthermore, what are some concrete examples of (computational exercises of) group actions?
Thanks.