Already got wrong chatgpt answerPlz all parts will upvote no chatgpt. Question 8The number of non-solvable groups of order 32 isQuestion 9The number of Sylow 3-subgroups of GL_2(3) isQuestion 10The number of Sylow 7-subgroups of S_7 is

Step 1: Question 1 Note that the prime factorization of 32 is 32=25, i.e., any group of order 32 is…

Answered: Question 8 The number of non-solvable…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.3: Zeros Of Polynomials

Problem 54E

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(5.2) Challenge Modern Applied Algebra. Show work.
let G={e, a, a^2, a^3, a^4, a^5, a^6, b, ab, a^2b, a^3b, a^4b, a^5b, a^6b}. related by the equation ba=a^2b, fined order of each element and proper nontrivial subgroups of G.
Are there any other subgroups of = order 3 & 2 ? For his vere'll consider all subsets of 3 and 2 elements of $3 and use equations of fiz and gf² = fg to check if they are subgroups 5-{e, f, f²g, fg, f²g] } All Subsets of 3 elements { e. f₁ f²} = = {e₁ he₁ f₁ g y = Not a Subgp. Since £g¢{ } 2 17 {e, f, f'g]=" { e, f ²₁ gb = {e, f²₁ fg} = 11 {e, 5², 3²g} = 11 11 " "1 11 11 " since f.f²g = fg-=9& { } Since 1²g & {y • Since f²fg. f²f g - f³ g - g 4 2 3 Since f²f²g = f g = fg &{} L> 71 Ee, g, fg} = not a subgp. since gffg)=ggf}=1² #23 Ee, g, fg}= Since g(8²) = g(gf) = f 483 E g, fg, fg}.. Since gf ggf) f €23 So the only subgroups of order. 3 are , 11 2 (1
A_4 in S_4 Find solutions to pic attached
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A group G has come to talk to you. You find out that G has 100 = 22 x 52 elements and that G has exactly one subgroup of order 10 whose name is Н. (a) Remind yourself, by rereading Example 10.7, why H must be a nor- mal subgroup of G. (b) Let PE Syl;(G). What can you say about |P|, |H^P|, and |(H, P)|? Prove your assertions. (c) The group G is guaranteed to have subgroups of which sizes?
give the order of the element in the factor group (2,0) + <(4,4)> in (Z6 x Z8)/<(4,4)>

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Question 8 The number of non-solvable groups of order 32 is Question 9 The number of Sylow 3-subgroups of GL_2(3) is Question 10 The number of Sylow 7-subgroups of S_7 is