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College Algebra Exam Review 141

# College Algebra Exam Review 141 - 151 3.1 DIRECT PRODUCTS Q...

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3.1. DIRECT PRODUCTS 151 Since the Q A i are mutually commuting subgroups, the subgroup gener- ated by any collection of them is their product: h Q A i 1 ; Q A i 2 : : : : ; Q A i s i D Q A i 1 Q A i 2 Q A i s : In fact, the product is f .x 1 ; x 2 ; : : : ; x n / 2 P W x j D e if j 62 f i 1 ; : : : ; i s gg : In particular, Q A 1 Q A 2 Q A n D P . Example 3.1.11. Suppose the local animal shelter houses several collec- tions of animals of different types: four African aardvarks, five Brazilian bears, seven Canadian canaries, three Dalmatian dogs, and two Ethiopian elephants, each collection in a different room of the shelter. Let A denote the group of permutations of the aardvarks, A Š S 4 . 1 Likewise, let B denote the group of permutations of the bears, B Š S 5 ; let C denote the group of permutations of the canaries, C Š S 7 ; let D denote the group of permutations of the dogs, D Š S 4 ; and finally, let E denote the group of permutations of the elephants, E Š S 2 Š Z 2 . The symmetry group of the entire zoo is P D A B C D E .
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