Question

In: Advanced Math

2. (a) Let p be a prime. Determine the number of elements of order p in...

2. (a) Let p be a prime. Determine the number of elements of order p in Zp^2 ⊕ Zp^2 .

(b) Determine the number of subgroups of of Zp^2 ⊕ Zp^2 which are isomorphic to Zp^2 .

Solutions

Expert Solution

Answer part A

page 2

Answer Part (b)

Page 2


Related Solutions

Let G be a group of order p am where p is a prime not dividing...
Let G be a group of order p am where p is a prime not dividing m. Show the following 1. Sylow p-subgroups of G exist; i.e. Sylp(G) 6= ∅. 2. If P ∈ Sylp(G) and Q is any p-subgroup of G, then there exists g ∈ G such that Q 6 gP g−1 ; i.e. Q is contained in some conjugate of P. In particular, any two Sylow p- subgroups of G are conjugate in G. 3. np ≡...
11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]p, [2]p, ....
11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]p, [2]p, . . . , [p-1]p}. Prove that for y ≠ 0, Ly restricts to a bijective map Ly|s : S → S. 11.5 Prove Fermat's Little Theorem
Let p be the prime number (2^20)*(3^7)5 + 1 = 11466178561. Solve for x such that...
Let p be the prime number (2^20)*(3^7)5 + 1 = 11466178561. Solve for x such that 2^x ≡ 2376886429 (mod p) Explain your method carefully.
Let p be an odd prime (i.e., any prime other than 2). Form two vector spaces...
Let p be an odd prime (i.e., any prime other than 2). Form two vector spaces V1, V2 over Fp (prime field of order p) with bases corresponding to the edges and faces of an icosahedron (so that V1 has dimension 30 and V2 has dimension 20). Let T : V1 → V2 be the linear transformation defined as follows: given a vector v ∈ V1, T(v) is the vector in V2 whose component corresponding to a given face is...
(a) Let G be a finite abelian group and p prime with p | | G...
(a) Let G be a finite abelian group and p prime with p | | G |. Show that there is only one p - Sylow subgroup of G. b) Find all p - Sylow subgroups of (Z2500, +)
Let gcd(a, p) = 1 with p a prime. Show that if a has at least...
Let gcd(a, p) = 1 with p a prime. Show that if a has at least one square root, then a has exactly 2 roots. [hint: look at generators or use x^2 = y^2 (mod p) and use the fact that ab = 0 (mod p) the one of a or b must be 0(why?) ]
Throughout this question, let G be a finite group, let p be a prime, and suppose...
Throughout this question, let G be a finite group, let p be a prime, and suppose that H ≤ G is such that [G : H] = p. Let G act on the set of left cosets of H in G by left multiplication (i.e., g · aH = (ga)H). Let K be the set of elements of G that fix every coset under this action; that is, K = {g ∈ G : (∀a ∈ G) g · aH...
let E be a finite extension of a field F of prime characteristic p, and let...
let E be a finite extension of a field F of prime characteristic p, and let K = F(Ep) be the subfield of E obtained from F by adjoining the pth powers of all elements of E. Show that F(Ep) consists of all finite linear combinations of elements in Ep with coefficients in F.
Let p be an integer other than 0, ±1. (a) Prove that p is prime if...
Let p be an integer other than 0, ±1. (a) Prove that p is prime if and only if it has the property that whenever r and s are integers such that p = rs, then either r = ±1 or s = ±1. (b) Prove that p is prime if and only if it has the property that whenever b and c are integers such that p | bc, then either p | b or p | c.
Let p be a prime and d a divisor of p-1. show that the d th...
Let p be a prime and d a divisor of p-1. show that the d th powers form a subgroup of U(Z/pZ) of order (p-1)/d. Calculate this subgroup for p=11, d=5; p=17,d=4 ;p=19,d=6
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT