Abstract algebra. Please show step by step PROBLEM 1.10 PROBLEM 1.10 1. QUOTIENT

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Abstract algebra. Please show step by step

PROBLEM 1.10

PROBLEM 1.10

1. QUOTIENT GROUPS AND NORMAL SUBGROUPSs Problem 1.1. Find all the normal subgroups of (Z6, +) Problem 1.2. Prove that the set of compler matrices Wher A* denotes co Tripler con;jugation of A, and N denotes the transpose of matrir A, is a subgroup of GL(n, C) Problem! 1.3. Denote by SU (n)U (n) n SL(n, C), and pTove that U(n)SU(n) ~ U(1) Problem 1.4. Identifying the group of rotations in two dimensions, S0 (2), to the unit circle in C and denoting by Unz eS2-1], prove that SO(2)/Un is a group Problem 1.5. Recall that An denotes the set of peTTnutations E S11 of signature sign()-1. Prove that Sn/An ~Z2 Problem 1.6. Let M(0, oo) denote the set of differentiable, strictly mono- tone functions M(0, oo) {f : (0,00) (0,00)If E Cl (0,00),J'(z)0, Vz E (0,00)} Prove that M(0, oo) is a group with respect to function composition, and that the set of strictly increasing functioTLs 1(0,00){f : (0,00) > (0,00) If Cl (0,00), f,(z) > 0, Vz e(0,00)} is a subgroup, I(0, ooM(0, oo) Problem 1.7. Prove that M(0, oo)/I(0, oo)Z2 Problem! 1.8. Prove that for any n 1, 2, , (R", t) is a group, where the operation is vector addition. Then prove that RniTn/R" R'n Problem 1.9. Denote by SU(n)U(n) n SL(n, C), and prove that U(n)SU(n) ~ U(1) Problem 1.10. Find all the normal subgroups of So(2)

Explanation / Answer

the given group G=Z6 under addition

order of G=6

as we know by lagrange theorem"order of a subgroup divide the order of a group"

hence possible order of subgroup of G=1,2,3 and 6

hence the subgroup will be=Z6 ,2Z6 ,3Z6, {0}

and as we know that group is abelian

[result "every subgroup of a abelian group is normal "]

so all four subgroups are normal

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