Group of automorphisms

Group of automorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 1 Group of automorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 2 Group of automorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 3 Group of automorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 4 Group of automorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 5 Group of automorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 6 Group of automorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 7 Group of automorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 8 Group actions Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 1 Stabilizer subgroups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 2 Orbit-Stabilizer theorem Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 3 Orbit-Stabilizer theorem Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 4 Groups of prime order Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 5 Cauchy’s theorem Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 6 Cauchy’s theorem Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 7 Sylow’s first theorem Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 8 Sylow’s first theorem Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 9 Final remarks Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 10 Left cosets Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 1 Left closets Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 2 Normalizer subgroup Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 3 Normal subgroup Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 4 Left closets Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 5 Examples of normal subgroups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 6 Kernel is normal Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 7 Factor groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 8 Factor groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 9 Factor groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 10 Factor groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 11 Fundamental theorem of group homomorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 12 Fundamental theorem of group homomorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 13 Fundamental theorem of group homomorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 14 Fundamental theorem of group homomorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 15 Fundamental theorem of group homomorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 16 Fundamental theorem of group homomorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 17 Fundamental theorem of group homomorphisms Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 18 Finite abelian groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 1 Finite abelian groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 2 Finite abelian groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 3 Finite abelian groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 4 Finite abelian groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 5 Finite abelian groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 6 Finite abelian groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 7 Finite abelian groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 8 Finite abelian groups Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 9 QUIZ 4, VERSION A, MATH103A, SUMMER 2021 1. Give a brief explanation for each one of the following problems. (a) (2 points) Find the order of (1, 2, 3, 4)(4, 6, 7)(5, 8, 9, 10) ∈ S10 . (b) (2 points) Find the order of Aut(Z21 ). (c) (3 points) Is there a finite group G of order 45 such that o(g) is a power of 3 for every g ∈ G? (d) (3 points) Is there an action of a group G of order 120 on a finite set X that has an orbit of size 7? 2. Let S 1 := {z ∈ C | |z| = 1}. We know that (S 1 , ·) is a subgroup of C× := C\{0} under multiplication. (a) (3 points) Prove that C× /S 1 ≃ R+ where R+ is the group of positive real numbers under multiplication. (b) (3 points) Prove that R/Z ≃ S 1 . (Hint. Let’s recall that S 1 = {eiθ | θ ∈ R}.) (c) (3 points) Let n be a positive integer and Mn := {z ∈ S 1 | z n = 1}. Prove that S 1 /Mn ≃ S 1 . (Hint. Think about f : S 1 → S 1 , f (z) := z n .) 3. Suppose |G| = pk and gcd(p, k) = 1. (a) (1 points) Prove that G has a subgroup of order p. (b) (3 points) Suppose P is a normal subgroup of order p and o(xP ) is a power of p (as an element of the factor group G/P ). Prove that xP = P . (c) (2 points) Suppose P is a normal subgroup of order p and o(x) is a power of p for some x ∈ G. Prove that x ∈ P . 4. (5 points) Find the standard form of Z18 × Z75 × Z20 . 1 QUIZ 4, VERSION B, MATH103A, SUMMER 2021 1. Give a brief explanation for each one of the following problems. (a) (2 points) Find the order of (1, 2, 3, 4)(4, 6)(5, 8) ∈ S8 . (b) (2 points) Find the order of Aut(Z15 ). (c) (3 points) Is there a finite group G of order 40 such that o(g) is a power of 2 for every g ∈ G? (d) (3 points) Suppose G is a finite group which acts on a finite set X via ∗. Prove that for every x ∈ X we have |G ∗ x| divides |G|. 2. Let S 1 := {z ∈ C | |z| = 1}. We know that (S 1 , ·) is a subgroup of C× := C\{0} under multiplication. (a) (3 points) Prove that C× /S 1 ≃ R+ where R+ is the group of positive real numbers under multiplication. (b) (3 points) Prove that R/Z ≃ S 1 . (Hint. Let’s recall that S 1 = {eiθ | θ ∈ R}.) (c) (3 points) Let n be a positive integer and Mn := {z ∈ S 1 | z n = 1}. Prove that S 1 /Mn ≃ S 1 . (Hint. Think about f : S 1 → S 1 , f (z) := z n .) 3. Suppose G is a finite group, H and K are two subgroups, and gcd(|H|, |K|) = 1. (a) (3 points) Prove that H ∩ K = {eG }. (b) (2 points) Suppose in addition that H and K are normal subgroups. Prove that for every h ∈ H and k ∈ K, hkh−1 k −1 ∈ H ∩ K. (c) (1 points) Suppose in addition that H and K are normal subgroups. Prove that for every h ∈ H, K ⊆ CG (h). 4. (5 points) Find the standard form of Z12 × Z45 × Z20 . 1
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