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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- Fall '08
- Algebra, Equivalence relation, srk, symmetric group Sn