Simple Groups

Recall that a group G is simple if it has no normal subgroupsMathworldPlanetmath except itself and {e}. Let G be a finite simple group and let p be a prime number.

(a) Suppose G has precisely k Sylow p-subgroupsMathworldPlanetmathPlanetmath with k>1. Show that G is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to a subgroup of the symmetric groupMathworldPlanetmathPlanetmath Sk.

(b) With the same hypothesisMathworldPlanetmath, show that G is isomorphic to a subgroup of the alternating groupMathworldPlanetmath Ak.

(c) Suppose G is a simple groupMathworldPlanetmathPlanetmath that is a proper subgroupMathworldPlanetmath of Ak and k5. Show that the index [Ak:G]k.

(d) Prove that if G is a group of order 120 then G is not a simple group. (Parts (b) and (c) may be helpful.)

Title Simple Groups
Canonical name SimpleGroups
Date of creation 2013-03-22 19:30:43
Last modified on 2013-03-22 19:30:43
Owner jac (4316)
Last modified by jac (4316)
Numerical id 6
Author jac (4316)
Classification msc 20B05