The mathematics itself might not require doing arithmetic, but human psychology forces us to approach the abstract via the concrete. A facility with arithmetic

The mathematics itself might not require doing arithmetic, but human psychology forces us to approach the abstract via the concrete. A facility with arithmetic helps with building a pile of concrete examples as a springboard for the leap to the abstract. For example, the usual example of a cyclic group is {0,1,2,3,4,5} with addition modulo 6. Maybe we start with 12 rather than 6 because it is familiar to old people used to analogue clocks. We end up with one example for each size. But we could also give {1,2,3,4,5,6} with multiplication modulo 7. Now the abstract idea of an isomorphism has a concrete example. [0->1, 1->3, 2->2, 3->6, 4->4, 5->5]. And we get a new perspective on generators. {0,1,2,3,4,5} has an obvious generator: 1. We are tempted to view 5 as an extra nuisance generator. But we notice that 6x6 = 36 = 5x7+1 means that 6x6 = 1 in the multiplicative group so 6 isn't a generator. And 2x2x2 = 8 = 7+1 == 1 mod 7, so 2 isn't a generator either. Viewed abstractly, product groups look dull and easy. The Klein Four group is a product C2 x C2, and has four elements {(0,0), (0,1), (1,0), (1,1)}. Obviously the only set of generators is {(0,1),(1,0)} and there are two automorphisms, the identity and the map that swaps (0,1) and (1,0). Those with a facility with arithmetic might play with {1,3,5,7} and multiplication modulo 8. Or {1,5,7,11} and multiplication modulo 12. But beware, you are venturing into dangerous territory. Which one of 3,5, and 7 is (0,1) and which is (1,0)? You are going to discover that the previous paragraph is wrong. (1,1) looks different from (0,1) and (1,0). It looks inferior and not a generator. But actually there are three minimal sets of generators {3,5}, {5,7}. and {3,7}. Playing with similar examples gives a second insight into what mathematicians mean by isomorphism. The first insight is that names don't matter: an isomorphism is what a ConLanger would call a relex = relexification. You invent a new language by inventing new words for the same old stuff. The second insight is that we build compound names, with internal structure. We have primitive names, 0 and 1. We build compound names (0,0), (0,1), (1,0), and (1,1). Then the mathematical concept of isomorphism treats our set of compound names as a mere dust of points, discarding the structure. An isomorphism can as easily map (0,1) to (1,1) as to (1,0) because they are just names, like 3,5,7. A loose notion of isomorphism as "same shape" can trip you up, because the structure of the names isn't part of the shape. Knowing your times tables well enough to play happily with simple examples can help learners appreciate the subtleties of advanced math.