Abstract Algebra: Theory and Applications · Problem 14.5.2

The Sylow Theorems — Thomas Judson

Mathematics — Abstract Algebra · fixed-sets-stabilizers

Problem

Compute all XgX_g and all GxG_x for each of the following permutation groups. (a) X={1,2,3}X = \{1, 2, 3\}, G=S3={(1),(1 2),(1 3),(2 3),(1 2 3),(1 3 2)}G = S_3 = \{(1),(1\ 2),(1\ 3),(2\ 3),(1\ 2\ 3),(1\ 3\ 2)\} (b) X={1,2,3,4,5,6}X = \{1, 2, 3, 4, 5, 6\}, G={(1),(1 2),(3 4 5),(3 5 4),(1 2)(3 4 5),(1 2)(3 5 4)}G = \{(1),(1\ 2),(3\ 4\ 5),(3\ 5\ 4),(1\ 2)(3\ 4\ 5),(1\ 2)(3\ 5\ 4)\} Hint. (a) X(1)={1,2,3}X_{(1)} = \{1,2,3\}, X(1 2)={3}X_{(1\ 2)} = \{3\}, X(1 3)={2}X_{(1\ 3)} = \{2\}, X(2 3)={1}X_{(2\ 3)} = \{1\}, X(1 2 3)=X(1 3 2)=∅X_{(1\ 2\ 3)} = X_{(1\ 3\ 2)} = \emptyset. G1={(1),(2 3)}G_1 = \{(1),(2\ 3)\}, G2={(1),(1 3)}G_2 = \{(1),(1\ 3)\}, G3={(1),(1 2)}G_3 = \{(1),(1\ 2)\}.

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