Suppose c is a conjugacy class in a с group G such that IC is finite but | c/#1. Then there exists an cement of in G that does g not commute with element of C. Show that hypothesis of any finiteness of IC | is necessary for this.

Q: Suppose that g and h induce the same inner automorphism of a group G. Prove that h-'g E Z(G).

A: Given: g , h induce the same inner automorphism of a group G. Prove that: h-1g∈ZG.

Q: 3. A ring are defined busing THREE (3) axioms that need to satisfy such as Abelian Group,…

A:

Q: Verify that (Z, O) is an infinite group, where Z is the set of integers and the binary operator O is…

A: According to the given information, it is required to verify that:

Q: An element is called a square if it can be expressed in the form b2for some b. Suppose that G is an…

A: Given: An element is called a square if it can be expressed in the form b2 for some b.G is an…

Q: Prove that a subgroup of a finite abelian group is abelian. Be careful when checking the required…

A:

Q: 8) a) Let A be the set of all real numbers z 0, 1, 2. Define the permutation f: A→A by the formula…

A:

Q: Consider the set of integers with the operation * given by a b=a -b. Which of the following are…

A:

Q: Give an example of a non-Abelian group that has exactly threeelementsof finite order.

A: Let G=Z3⊕Z be an additive group where e=0,0. Let x∈G then it can be observed that x=a,b where a∈Z3,…

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

A: |G| = pqr.   Without loss of generality, let p<q<r. We claim two things :  (i) There exists a…

Q: Define on R the operation * by x*y = X+y+k, for all x,y element of R and k is fixed real number. The…

A: We have to check

Q: Show that the order of the group of  automorphism of a finite group of  order n is a divisor of n !.

A: we have to show that the order of the group of automorphism of a finite group of order n is a…

Q: Consider the group G = {x € R such that x* 0} under the binary operation x*y=-

A:

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: Show that there is no simple group of order pqr, where p, q, and rare primes ( p, q, and r need not…

A: We know the definition of simple group, A simple group is a group whose only normal subgroups are…

Q: Show that conjugacy is an equivalence relation on a group.

A:

Q: Show that a group of order pn, where p is prime, is solvable.

A: Let G be a group of order p^n, where p is a prime number.   We will prove by induction on n that G…

Q: Show that the order of the group of automorphism of a finite group of order n is a divisor of n!.

A: we have to show that the order of the group of automorphism of a finite group of order n is a…

Q: If G is a group, prove that Aut(G) and Inn(G) are groups

A:

Q: et f be an isomorphism from a grou onto a Group H. et a be an element in G and the orde Fa is 5. it…

A: We have to choose the correct answer from the given option according to the given condition

Q: True or false, The nonzero elements of ℚ form a cyclic group ℚ* under field multiplication.

A: Cyclic group under Multiplication

Q: Let G be a group (not ncesssarily an Abelian group) of order 425. Prove that G must have an element…

A:

Q: B Suppose (G, *) is group of order > 1. Which statement best describes problem? Cannot have zero,…

A:

Q: Without using the Generalized Cayley Theorem or its corollaries,prove that there is no simple group…

A: A simple group is a group with no non-trivial proper normal subgroups. We need to show that there is…

Q: 9 Find all isamorphism classes of abelian groups of arder 72.

A: We have find isomorphism classes of abelian group of order 72:

Q: The population of a community is known to increase at a rate proportional to the number of people…

A:

Q: Consider a group G with operation * and identity element e. Show that for any element a in G, there…

A: In any group G, we define a group as a set G equipped with an operation * that combines any two…

Q: G is a finite group and f be an automonphism of G satisfying the conditian f(x)= x → x=e for xEG.…

A:

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

A:

Q: Prove that there are exactly five groups with eight elements, up to isomorphism.

A:

Q: Show that for an abelian group, the only inner automorphism is the identity mapping, whereas for…

A: We have to show that for an abelian group, the only inner automorphism is the identity mapping,…

Q: Show that the quotient group Q/Z is isomorphic to the direct sum of prufer group

A:

Q: not

A: