3.1. DIRECT PRODUCTS151Since theQAiare mutually commuting subgroups, the subgroup gener-ated by any collection of them is their product:hQAi1;QAi2: : : : ;QAisi DQAi1QAi2QAis:In fact, the product isf.x1; x2; : : : ; xn/2PWxjDeifj62 fi1; : : : ; isgg:In particular,QA1QA2QAnDP.Example 3.1.11.Suppose the local animal shelter houses several collec-tions of animals of different types: four African aardvarks, five Brazilianbears, seven Canadian canaries, three Dalmatian dogs, and two Ethiopianelephants, each collection in a different room of the shelter.LetAdenote the group of permutations of the aardvarks,AŠS4.1Likewise, letBdenote the group of permutations of the bears,BŠS5; letCdenote the group of permutations of the canaries,CŠS7; letDdenotethe group of permutations of the dogs,DŠS4; and finally, letEdenotethe group of permutations of the elephants,EŠS2ŠZ2. The symmetrygroup of the entire zoo isPDABCDE.
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