ANSWER:
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...
Let G be a finite group such that p is a prime and p divides |G|. Let P be a p-Sylow subgroup of G such that P is cyclic and ? . Let H be a subgroup of P . Prove We were unable to transcribe this imageWe were unable to transcribe this image
ame: . (10 points) Let p > 3 be any prime number. (a) Show that p mod 6 is equal to 1 or 5 (b) Use part (a) to prove that pe - 1 is always a multiple of 24.
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrutic residue modulo pif and only ifp-a is a quadratic non-residue modulo p. 7.24 Theorem. Let p be a prime of the form p odd prime. Then p 3 (mod 4). 241 where q is an The next theorem describes the symmetry between primitive roots and quadratic residues for primes arising from odd Sophie...
(1) Let p be a prime number. Describe all the groups with p elements. (2) Let # be a permutation in S(4). What are the possible orders of T according to Lagrange's theorem? (3) Show that there are no elements of order 8 in S(4) (even though 8 divides 24 = 4!).
10. Let [n] be an element in Zp, p prime. We say [n] is perfect provided [o (n)] [2n]. Show that d-[21,where Idy]-'is the multiplicative inverse of ld in Z, [dkl In 10. Let [n] be an element in Zp, p prime. We say [n] is perfect provided [o (n)] [2n]. Show that d-[21,where Idy]-'is the multiplicative inverse of ld in Z, [dkl In
2. Let p be an odd prime. We saw last week that the problem of counting solutions to the congruence (mod p) is only interesting when p has the form 4k1. For the rest of this problem let p 4k+1. (a) Show that (mod -1 5 (mod 8) (b) Show that pEl (mod 8 -1 p5 (mod 8) (c) Draw condlusions about the number of solutions to these congruences 14 (mod p) -1 (mod p) (mod p) 2. Let p...
Write your own answers 10. Let p be a prime number, and let a be an integer that is not divisible by p. Prove that the congruence equation ax = 1 mod p has a solution X e Z.
(4) (10 points) Show that 3 is a prime element in Zg]. Find the irreducible Z8]. Specify the irreducible factors that appear in the factorization of 9t. ation of 9i in let Prime P-(3) Thus 3 divide s Nea) ar 3 divides NC) ud we can sahat divides N(O)x possibility t, Hat bof ,, hak rosidue omod 3 Henle 9ie3 and p.3 (s prime. 3, us Sethat we are left wl oor 1 , So the only if you work...
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The relation T is a partial order relation (you do not need to prove this). Draw the Hasse diagram for T 1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the...
Let p be a prime. If for integers k and I we have rk = x (mod p) for all x E Z, (x,p) = 1 show that k =l (mod p – 1).