The number of Sylow p-subgroups in G, for a given prime p, is of the form 1+kp, where   k is some non-negative integer and (1+kp)|o(G

Q: m and n be positive integers so that m n. Show for any integer a that the congruence

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Q: Let K be a finite field of order q = p", with p an odd prime number and n > 1. We denote by Kײ the…

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Q: If the order of a group G is p²q² with p and q as distinct primes, then G is not simple.

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Q: Assume that (G, ) is a group, that geG, and that t is the smallest positive integer such that g' =…

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Q: Suppose that G is a finite simple group and contains subgroups Hand K such that |G:H| and |G:K| are…

A: Given : G is a finite simple group and contains subgroups H and K such that G:H and G:K are prime.…

Q: If n is not prime, then G = {1, 2, 3,..., n-1} is not a group under multiplication mod n.

A: We prove this result by contradiction.  Let n is not prime and G={1,2,3,....,n-1} is a group under…

Q: You may have noticed that all the “hard integers” are even. Choosethree odd integers between 200 and…

A: In this question, we have to choose any three odd integers between 200 and 1000 and show that none…

Q: The index of [G:H] is a prime number that divides H and divides G O the double number of left or…

A: The double number of left or right cosets of H in G.

Q: Let a and b belong to some group. Suppose that |a| =  m, |b| = n,and m and n are relatively prime.…

A: Given: A group G and a, b∈G. Also, a=m, b=n and m and n are relatively prime. To Do: If ak=bk for…

Q: et H ≤ S4 be the subgroup consisting of all permutations σ that satisfy σ(1) = 1. Find at least 4…

A: This is a good exercise in working with cosets. We first find out the subgroup $H$ and then working…

Q: (a) G = Z12 and H = { 0, 2, 4, 6, 8, 10 } (based from the operation table) (b) G = Z; and H = { 1,…

A: Given a G=Z12 and H=0,,2,4,6,8,10b G=Z7* and H=1,2,4c G=Z* and H=a+b2 : a,b∈Q and a,b are not both…

Q: f the order of a group G is p2q2 with p and q as distinct primes, then G is not simple.

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Q: Show that every element of GF(pn) can be written in the form ap forsome unique a in GF( pn).

A: In this question, we have to show that, every element of GF(pn) can be written in the form ap for…

Q: list the elements of the subgroup generated by the subet {2,3} of Z12

A: Let

Q: 4 a

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Q: ] Let N be the subgroup of (Z15, +) generated by 3, that is N = . i- What is the order of N+ 2 in…

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Q: . Find all of the conjugacy classes of elements of D14

A: Note: The notation of a dihedral group in different books varry from Dn to D2n. Here I consider it…

Q: Let s in G, s is not equal to 1.

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Q: Find all the cosets of the subgroup 4Z of Z

A: In the group of integers under addition, the subgroup consists of all multiples of 4, i.e., .A coset…

Q: 9. Find all proper nontrivial subgroups of Z₂ × 2₂.

A: The given group is: .To Do: We have to find all proper nontrivial subgroups of .Note: Since you have…

Q: Let G be a finite group and p be an odd prime number. Show that G can not have exactly p elements of…

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Q: If H is normal, then xHxH KCH. Since o(K)= o(H), then we obtain H = K, this shows that H is unique.…

A: Suppose H be  the unique sylow P-subgroup  of G.For all  g∈G,g-1Hg  is  a  sylow p-subgroup  and by…

Q: et n ≥ 3. (a) Find the number of permutations w ∈ Sn such that 1, 2, 3 belong to three distinct…

A: denotes symmetric group.

Q: Let G be a finite abelian group. Then there exists a finite sequence of positive integers (pa¹,…

A: Given : Let G be a finite abelian group.To Prove :Then there exists a finite sequence of positive…

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A: The solution is given as

Q: Show that every element of GF(pn) can be written in the form ap forsome unique a in GF( pn).

A: Given: GFpn. To show: Every element of GFpn can be written in the form ap for some unique a in GFpn.

Q: How many Sylow 5-subgroups of S5 are there? Exhibit two.

A: We have to find Sylow 5-subgroups of S_5

Q: Let P be a Sylow 19-subgroup and Q be a Sylow 7-subgroup. Then PQ is a subgroup of G of order: O 21…

A: We have given that ,  P be a Sylow 19-subgroup. Q be a Sylow 7-subgroup. We need to find , order of…

Q: Let G = Z*20, the set of units in Z20 with multiplication as operation, and let g = 3 ∈ Z*20. What…

A: The objective of the question is to find the elements of the group G = Z*20 and the subgroup H =…

Q: Determine the number of Sylow 2-subgroups of D2m, where m is anodd integer at least 3.

A: As per the question we are given the dihedral group D2m and we have to find the number of total…

Q: In Z, find all generators of the subgroup <m> <n>.

A: The objective of the question is to find all generators of the subgroup <m> <n> in the…

Q: Let G be a finite Abelian group and let p be a prime that divides the order of G. Then G has an…

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