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College Algebra Exam Review 235

# College Algebra Exam Review 235 - that can be made from...

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5.2. GROUP ACTIONS—COUNTING ORBITS 245 We can count F in two different ways: j F j D X x 2 X X g 2 G 1 F .x; g/ D X x 2 X j Stab .x/ j and j F j D X g 2 G X x 2 X 1 F .x; g/ D X g 2 G j Fix .g/ j : Dividing by j G j , we get 1 j G j X g 2 G j Fix .g/ j D X x 2 X j Stab .x/ j j G j D X x 2 X 1 j O .x/ j : The last sum can be decomposed into a double sum: X x 2 X 1 j O .x/ j D X O X x 2 O 1 j O j ; where the outer sum is over distinct orbits. But X O X x 2 O 1 j O j D X O 1 j O j X x 2 O 1 D X O 1; which is the number of orbits! Thus, we have the following result, known as Burnside’s lemma. Proposition 5.2.2. (Burnside’s lemma). Let a finite group G act on a finite set X . Then the number of orbits of the action is 1 j G j X g 2 G j Fix .g/ j : Example 5.2.3. Let’s use this result to calculate the number of necklaces
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Unformatted text preview: that can be made from four red beads, three white beads, and two yellow beads. X is the set of 9Š 4Š3Š2Š D 1260 arrangements of the beads, which we locate at the nine vertices of a nonagon. Let g be an element of D 9 and consider the orbits of h g i acting on vertices of the nonagon. An arrangement of the colored beads is ﬁxed by g if, and only if, all vertices of each orbit of the action of h g i are of the same color. Every arrangement is ﬁxed by e ....
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