Let H be a normal subgroup of a finite group G and let a be an elementof G. Complete the following statement: The order of the elementaH in the factor group G/H is the smallest positive integer nsuch that an is .

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: Suppose o is a homomorphism of groups fom G onto H. If H has an element of order 100, then show that…

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Q: Prove that there is no simple group of order p2q, where p and q areodd primes and q > p.

A: Let G be a group. |G| = p2q, where p and q are odd primes and q > p. The first Sylow Theorem: A…

Q: Prove the next statement: If G is an abelian group of finite order n, then G has a subgroup of…

A: Answer

Q: Prove O3 is not a group

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Q: Prove that for a fixed value of n, the set Un of all nth roots of 1 forms a group with respect to…

A: Fix an n∈ℕ. Let U=u∈ℂ | un=1 .  Note that U⊂ℂ* and ℂ* is a group under multiplication. Let u,v∈U…

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: In Z, find all generators of the subgroup <10>  <12>.

A: The objective of the question is to find all generators of the subgroup in the set of integers…

Q: Let f be the following permutation: 1 2 3 4 5 6 7 f = 4 3 6 2 1 5 7 a. Determine the group to which…

A: Given permutation is f=12345674362157 f is permutation of 7 elements so it is belong to s7. i.e. sn…

Q: Let H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

A: Given that H and K are the finite subgroups of a group G and also an element a such that a∈G Here,…

Q: 1. Let p be a prime. A group G (either finite or infinite) is called p-group if the order of every…

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Q: 22, Use mathematical induction to prove that if a1, a2, ... , an are elements of a group G, then…

A: See the detailed solution below.

Q: If G is a group and a1, a2,…, an shows that a1 * a2 *… * an is unique, regardless of the order in…

A: We are given that G is a group. (G satisfies all the group axioms). Suppose * is the defined binary…

Q: 5. Define the right regular action of a group G on itself. Show it is a group action. Is it…

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Q: Let m be a positive integer. If m is not a prime, prove that the set {1,2,..., m – 1} is not a group…

A: We show that it doesn't satisfy clousre property.

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: (d) Define * on Q by a * b = ab. Determine whether the binary operation * gives a group on a given…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Let H be a proper subgroup of a finite group G. Show that G is notthe union of all conjugates of H.

A: Let H be a proper subgroup of a finite group G. We how prove that G is not the union of all…

Q: @1: Let G be a finite group and aEG 1.t la1= 12. If H= <a) find all other generalrs of H.

A: Let G be a finite group and a∈G such that a=12. If H=a, find all other generators of H. H is a…

Q: Let G={aeR: a>0}. Select all of the operations from the following list such that (G, ◊) is a group.…

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Q: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.

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Q: Suppose that A is subset of a group G such that              A = {a : a2 =e  , a in G }    Where e…

A: Sol:- The subset A consists of all elements a in the group G such that a squared is equal to the…

Q: Let (G, *) and (H, *) be groups. The direct product of G and H is the set G × H equipped with…

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Q: Let G = U(32) and H = {1, 15}. The group GIH is isomorphic to one of Zg, Z, O Z, or Z, O Z, O Z,.…

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Q: For a group G and a, b € G, show that (aba-¹)= aba-¹ for any positive integer

A: The given statement,For a group  and , then  for any integer.The aim is to show that the statement…

Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

Q: Any product of solvable groups is solvable.

A: Prove that, any product of solvable group is solvable.

Q: Show that the order of each element of the quotient group Q/Z is finite.

A: We have to show that the order of each element of the quotient group Q/Z is finite.

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 in a finite group every element is conjugate to its inverse, then show that all the entries in…

A: Let G be a finite group and g∈G.We have to show that if g and g-1 are conjugate then χ(g) is real…

Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: Use the three Sylow Theorems to prove that no group of order 45 is simple.

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Q: 19. Let H and K be subgroups of a group G. Prove that gH ngK is a coset of HOK in G.

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Q: Show that a group G order pqr for distinct primes p ,q and r ,is solvable

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