(i) Determine whether φ defines a homomorphism. (ii) Find ker ф :-(g E G I ф(G)-e) and inn ф d(G). (ii) Draw Cayley...
(i) Determine whether ф defines a hornom Orphism. (ii) Find ker ф :-0€ G | ф(G) e} and in ф ф(G). (ii) Draw Cayley diagrams of the domain and codomain, and arrange them so one can "visually see" the cosets of ker ф in G. Draw dotted lines around these cosets. (iv) Is the quotient G/kero a group? If so, what is it isomorphic to? Z, defined by ф(n) n (mod 3). Here is an example of Step (iii) for...
Answer Question 5 . Name: 1. Prove that if N is a subgroup of index 2 in a group G, then N is normal in G 2. Let N < SI consists of all those permutations ơ such that o(4)-4. Is N nonnal in sa? 3. Let G be a finite group and H a subgroup of G of order . If H is the only subgroup of G of order n, then is normal in G 4. Let G...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...
I have to use the following theorems to determine whether or not it is possible for the given orders to be simple. Theorem 1: |G|=1 or prime, then it is simple. Theorem 2: If |G| = (2 times an odd integer), the G is not simple. Theorem 3: n is an element of positive integers, n is not prime, p is prime, and p|n. If 1 is the only divisor of n that is congruent to 1 (mod p) then...