Congruences : Primes, Inverse Modulo, GCD and Wilson's theorem

Please assist me with the attached congruence problems (hint: use Wilson's Theorem) a)Prove if a,b,c Z, N and gcd (c, ) = , then ac bc(mod ) if and only if a b (mod ). b) Let a Z, N, and p > 2 be a prime. Prove that a is its own inverse modulo p if and only if a 1 (mod p ). C) Let a,b Z, N .prove that ax ...continues

Group theory proofs

a. Let =2 +1 (2 (Power 2(power n))) Plus 1. Prove that P is a prime Dividing , then the smallest m such that P (2 -1) is m = 2 (hint use the Division Algorithm and Binomial Theorem) Please see attached.

Theory

Suppose that... Use Lagrange's Theorem Please see attached.

Cyclic Groups, Generators and Orders of Elements

(1) Let G be a group such that ... Show that G cannot be cyclic. (2) Show that a cyclic group with one generator has at most 2 elements. (3) Let a 2 G be an element of order two, and b 2 G an element of order three. Show that HK where H = (a) and K = (b) has order 6.

Orders of Cyclic Groups, Prime Order and Subgroups

(1) Let G be cyclic of order pn, p prime. Let H,K < G. Show that either H C K or K C H. (2) Let G 6 ≠ {e} such that it has no proper subgroups. Then G must be cyclic of prime order. (3) If G is a group with order pq where p > q are primes and q does not divide p − 1, then G must be cyclic. Please see the att ...continues

Superincreasing Sequence and Prove that Expression is Prime

1) Let S= {b ,b ,………..b } satisfies b j+1 >2b j for all j =1,2,3,…….n-1. Prove that S is a superincreasing sequence. ≡ 2) Prove that n E N with n ≥ 3 is prime if and only if there exists an integer m such that m^(n-1) ≡ 1 (mod n)^((n-1)/q) but m is not ≡ ( mod n) for any prime q| (n-1)

A problem related to period(order) of an element in a group

Let a and x be elements in a group G. Prove that a and axb ,where b is the inverse of a, have the same period.

Groups, Subgroups and Containment

Let H and K be subgroups of a group G such that one is not contained in the other. Prove that H U K is not a subgroup of G.

Irreducible and one-dimensional representations

1. Classify irreducible representations of Z over C. 2. Classify one-dimensional representations of Sn over any field k such that char k is not equal to 2.

Irreducible representations

1. Assume that the field is algebraically closed and has zero characteristic, G is finite and representations are finite-dimensional. Show that this statement is true under the above assumptions: "Let p be an irreducible representation of G, and q be an irreducible representation of H. Is it always true that the exterior t ...continues