If in a finite group every element is conjugate to its inverse, then show that all the entries in the character table are real.

Q: THE SET (1, 2, 4, 7, 8, 11, 13, 14) IS A GROUP UNDER MULTIPLICATION MODULO 15. THE INVERSES OF 4 AND…

A:

Q: Give all the possible elementary divisors of a group of order 40.

A: 40=23×5 So, the possible elementary divisors of the group are 2,2,2,5,2,4,5 and 8,5

Q: Please write if the set that is given is a semigroup or a monoid or a group of none of all wi…

A:

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: Give an example: The product of two solvable groups need not to be solvable?

A: it is clear that s2 is solvable because it is abelian

Q: Please help me by providing details definitions and theorems   prove that no group of order of 160…

A:

Q: Suppose that Z10 and Z15 are both homomorphic images of a finitegroup G. What can be said about |G|?…

A: Given: Z10 and Z15 are both homomorphic images of a finite group G. We have to tell about G and…

Q: 3) Determine whether each of the following is or is not a group: a) G = {m e Z|m is odd }, with…

A: Here G is not a group as it fails to satisfy the multiplicative inverse property.

Q: f(x) = -12-21x + 18x^2 - 2.75x^3 what method will give a root of the function equal to - 0.287425 in…

A: We will use a numerical method to determine what will be the value of root of the given function in…

Q: The inverse of each element in the group ({1, 3, 5, 7}, ∙8)

A:

Q: Every element in a ring has an Multiplicative inverse?

A: A set R is said to be ring with two binary operations addition and multiplication, if R is…

Q: Show that Ø is a ring homomorphism.

A:

Q: The set of non-zero real numbers with arithmetic multiplication as an operation is a mono but not a…

A: This statement is false. Because the set of non zero real no is a group.

Q: A ring may have more than one identity element for multiplication? T or F

A:

Q: Prove that if a, b ∈ G (group) and ab = ba, then |ab|  is a divisor of the least common multiple of…

A:

Q: The number of zero divisors in the ring Z80

A: Step:-1 Theorem:- If  gcda, n>1  and n- a then [a] is a zero divisor in ℤn. Also, we know ϕn =…

Q: Prove roots of unity are a subgroup

A:

Q: Determine the behaviour of the set of 4 elements {1,-1,i,-i} under the ordinary multiplication of…

A: Given, The set of 4 elements are 1,-1,i,-i.It is to be shown that this set is group.

Q: write why, in order to describe the homomorphic images of a group, it suffices to describe the…

A: The homomorphic images of a group refer to the set of all possible images that result from applying…

Q: Write Q(sqrt2) for the set described in exercise 3.1 (iii). given a non zero element a + bsqrt(2) of…

A: It refers to the set Q(2), which is the set of all numbers of the form a+b2, where a and b are…

Q: The product of 16 and 7.5 belongs to what group in the Real Number System?

A: Assuming the product means multiplication. (R\{0},*) forms group under multiplication. Also,…

Q: Explain why the ring of integers Z under usual addition and multiplication is not a field.

A: Given that the set of integers Z is a ring under addition and multiplication. We know that an…

Q: Write the three-dimensional matrices that represent each of the operations of the point group D2n…

A: The dihedral group D2n is defined by the collection of symmetries of a n-sided regular polygon. So…

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

A:

Q: The number of zero divisors of the ring Z4 O Z, is

A:

Q: Find the decomposition of the representation Sym³V

A: The objective is to find the decomposition of the representation of Sym3V

Q: Give an example of a noncommutative ring with unity.

A:

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

A:

Q: Let H be the subset of C consisting of the nth roots of unity. Prove that H is a group under…

A: