18
the operation fis associative by determining if the final six erualities of Example 18 holed defined in Example 15 on the set John, Sue, Henry, Pam,detr mine1 Determine if the operation defined in Example 1.5 is commutative, has an identity, and if each element has an inverse. Section 1.3 Groups rite out the Cayley table for the group(Zo to) and identify the inverse of enelh he inverse of ench element erind the 10 elements of the group z, x Z2 and write out the Cayley table. Itecall thnt its operation uses +s in the first coordinate and +2 in the second. Identity the inverse of each element. does Theorem 1.14 tell us about the entries in the Cayley table of a group? e that a set with exactly one element, A = {a), will always form a group since there one way to define an operation on the set. Be sure to show that the properties of a group all hold. even create groups with games! Consider four cups placed in a square pattern 21. W table. If we have a penny in one of the cups there are four ways we can move it to on a r cup: Horizontally, Vertically, Diagonally, or Stay where it is. We will label them D. S. To define an operation, consider two movements in a row,i.e., y means we anothe example, we and the inverse of each elemont ve the penny as a tells us to, then after that move the penny as y instructs. For le, H*V D since if we first move it horizontally and then vertically altogether e moved it diagonally. Create the Cayley table for this T