PROBLEM 1 Recall that there is a group homomorphism ε : Sn → {±1} which has the property that (y) = (−1) +1 for any k-cycle y. The kernel Ker(e) is denoted An. Now let л,σE S7 be the permutations πT = 1 2 3 4 5 6 7 64527 13 and = 0 (1234)(2 5 3 6) (27). 1.1 Find the cycle decompositions and orders of 7 and σ. 1.2 Find the values ε(T), ε(σ) and ε(70). Use this to determine which of the permutations π, σ and σ belong to A7. 1.3 Use Lagrange's theorem to find the order of An, for any n € Z+.

PROBLEM 1 Recall that there is a group homomorphism ε : Sn → {±1} which has the property that (y) = (−1) +1 for any k-cycle y. The kernel Ker(e) is denoted An. Now let л,σE S7 be the permutations πT = 1 2 3 4 5 6 7 64527 13 and = 0 (1234)(2 5 3 6) (27). 1.1 Find the cycle decompositions and orders of 7 and σ. 1.2 Find the values ε(T), ε(σ) and ε(70). Use this to determine which of the permutations π, σ and σ belong to A7. 1.3 Use Lagrange's theorem to find the order of An, for any n € Z+.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.7: Direct Sums (optional)

Problem 1E: Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under...

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