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Unformatted text preview: 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 S 4 . 1 Likewise, let B denote the group of permutations of the bears, B S 5 ; let C denote the group of permutations of the canaries, C S 7 ; let D denote the group of permutations of the dogs, D S 4 ; and finally, let E denote the group of permutations of the elephants, E S 2 Z 2 . The symmetry group of the entire zoo is...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08