if any confusion please comment...
(1.) Show that the ideals in Zm are precisely the set of the form < d> where d divides m
1. Show that the set of rational numbers of the form m /n, where m, n E Z and n is odd is a subgroup of QQ under addition. 2. Let H, K be subgroups of a group G. Prove: H n K is a subgroup of G 3. Let G be an abelian group. Let S-aEG o(a) is finite . Show that S is a subgroup of G 4. What is the largest order of a permutation in S10?...
3. If the integers mi, i = 1,..., n, are relatively prime in pairs, and a1,..., an are arbitrary integers, show that there is an integer a such that a = ai mod mi for all i, and that any two such integers are congruent modulo mi ... mn. 4. If the integers mi, i = 1,..., n, are relatively prime in pairs and m = mi...mn, show that there is a ring isomorphism between Zm and the direct product...
1. Use mathematical induction to prove ZM-1), in Ik + 6 for integers n and k where 1 <k<n - 1. = 2. Show that I" - P(m + k,m) = P(m+n,m+1) (m + 1) F. (You may use any of the formulas (1) through (14”).)
QUESTION 19 Let P(m, n) be the statement "m divides n", where the domain for both variables consists of all positive integers. (By “m divides n” we mean that n = km for some integer k.). is an Vm P(m,n). O a. False b. "False" and "not a tautology" O c. True d. Not a tautology QUESTION 23 Let P(m, n) be the statement "m divides n", where the domain for both variables consists of all positive integers. (By “m...
10. Let p be a prime number. We know that p divides (p- 1)!+1. Show that if p> 5 then (p- 1)!+1 is never of the form pë where e e Z0
10. Let p be a prime number. We know that p divides (p- 1)!+1. Show that if p> 5 then (p- 1)!+1 is never of the form pë where e e Z0
4. Let M be the set of 2 x 2 matrices of the form (62) where a, d E R - {0}. Consider the usual matrix multiplication ·, i.e: ae + bg af + bh ce + dg cf + dh (a) Show that (M,·) is an abelian group. 1 (b) Compute the cyclic subgroup generated by M = What is the order of M? 66 -4) (1) EM EM.
Let M be the set of 2 x 2 matrices of the form (82) where a, d ER-{0}. Consider the usual matrix multiplication, i.e: ae + bg af +bh ce + dg cf + dh (2)) = (ce ) (a) Show that (M,-) is an abelian group. (b) Compute the cyclic subgroup generated by M = What is the order of M? (6 -4) € M.
Show that the set of matrices of the form
where a, b ∈ Q is a field under the operations of matrix addition
and multiplication. (abstract algebra)
please show the following axioms (closure, identity,
associative, distributive, inverse, and commutative) for addition
and multiplication
a 6 26 a
Show that set of all vectors of the form (a, b, c, d) of R4 such that a = b + c + d is subspace of R4, whereas the set of all vectors of the form (a, b, c, d) of R4 such that a = b + c + d + 2 is not subspace of R4
Abstract Algebra
(1) Let I, J C R be ideals. Show that if I is generated by n elements, and J is generated by m elements, then I +J is generated by no more than nm elements. 1