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College Algebra Exam Review 232

College Algebra Exam Review 232 - 11Š 2Š4Š4Š D 34650...

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242 5. ACTIONS OF GROUPS n Example 5.1.20. The number of ordered sequences of k items chosen from a set with n elements is .n k/Š : Proof. The proof is similar to that for the previous example. This time let S n act on the set of ordered sequences of k -elements from f 1; 2; : : : ; n g . n Example 5.1.21. The number of sequences of r 1 1 ’s, r 2 2 ’s, and so forth, up to r k k ’s, is .r 1 C r 2 C C r k r 1 Šr 2 Š : : : r k Š : Proof. Let n D r 1 C r 2 C C r k . S n acts transitively on sequences of r 1 1 ’s, r 2 2 ’s, : : : , and r k k ’s. The stabilizer of .1; : : : ; 1; 2; : : : ; 2; : : : ; k; : : : ; k/ with r 1 consecutive 1 ’s, r 2 consecutive 2 ’s, and so on is S r 1 S r 2 S r k : n Example 5.1.22. How many distinct arrangements are there of the letters of the word MISSISSIPPI? There are 4 I’s, 4 S’s, 2 P’s and 1 M in the word, so the number of arrangements of the letters is
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Unformatted text preview: 11Š 2Š4Š4Š D 34650 . Exercises 5.1 5.1.1. Let the group G act on a set X . Define a relation on X by x ´ y if, and only if, there is a g 2 G such that gx D y . Show that this is an equivalence relation on X , and the orbit (equivalence class) of x 2 X is Gx D f gx W g 2 G g . 5.1.2. Verify all the assertions made in Example 5.1.4 . 5.1.3. The symmetric group S n acts naturally on the set f 1;2;:::;n g . Let ± 2 S n . Show that the cycle decomposition of ± can be recovered by con-sidering the orbits of the action of the cyclic subgroup h ± i on f 1;2;:::;n g . 5.1.4. Verify the assertions made in Example 5.1.6 ....
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