11. He also made fundamental contributions to the representation theory of the symmetric and alternating groups . 12. They alternate groups of stiff, shallow wingbeats with glides. 13. All perfect alternating groups have perfect double covers. 14. Arrange twigs into groups and glue them onto frame using glue gun, alternating groups lengthwise and crosswise. 15. For a concrete example, the alternating group on four points, A _ 4, is solvable but not supersolvable. 16. Alternating groups sometimes behave as if they were groups of Lie type over the field with one element.17. Analogously, the alternating group is a subgroup of index 2 in the symmetric group on " n " letters. 18. And in fact, the smallest simple non-cyclic group is " A " 5, the alternating group over 5 elements. 19. This has a center of order 2; the quotient modulo the center is isomorphic to the alternating group A 8. 20. There is a group of judges and an alternate group called the School of Hard Knocks, which judges each contestant.
