(a) Let G and G′ be finite groups whose orders have no common factors. Show that...

(a) Let G and G′ be finite groups whose orders have no common factors. Show that the only homomorphism φ:G→G′ is the trivial one.

(b) Give an example of a nontrivial homomorphism φ for the given groups, if an example exists. If no such homomorphism exists, explain why.

i.φ: Z16→Z7

ii.φ: S4→S5

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Show that the groups of the following orders have a normal Sylow subgroup. (a) |G| =...

Show that the groups of the following orders have a normal Sylow subgroup. (a) |G| = pq where p and q are primes. (b) |G| = paq where p and q are primes and q < p. (c) |G| = 4p where p is a prime greater than four.

Let G be a finite group and H a subgroup of G. Let a be an...

Let G be a finite group and H a subgroup of G. Let a be an element of G and aH = {ah : h is an element of H} be a left coset of H. If b is an element of G as well and the intersection of aH bH is non-empty then aH and bH contain the same number of elements in G. Thus conclude that the number of elements in H, o(H), divides the number of elements...

Let g(x) be a function so that the following covariance and expectations are finite numbers. Show...

Let g(x) be a function so that the following covariance and expectations are finite numbers. Show that cov [g(U), g(1 − U)] = (1/2) E {[g(U1) − g(U2)][g(1 − U1) − g(1 − U2)]} , where U ∼ U(0, 1), U1 and U2 are iid U(0, 1). Note you need the fact about the independences of U1 & 1 − U2 as well as 1 − U1 & U2 to show the above identity.

Let a be an element of a finite group G. The order of a is the...

Let a be an element of a finite group G. The order of a is the least power k such that ak = e. Find the orders of following elements in S5 a. (1 2 3 ) b. (1 3 2 4) c. (2 3) (1 4) d. (1 2) (3 5 4)

Let (G,·) be a finite group, and let S be a set with the same cardinality...

Let (G,·) be a finite group, and let S be a set with the same cardinality as G. Then there is a bijection μ:S→G . We can give a group structure to S by defining a binary operation *on S, as follows. For x,y∈ S, define x*y=z where z∈S such that μ(z) = g_{1}·g_{2}, where μ(x)=g_{1} and μ(y)=g_{2}. First prove that (S,*) is a group. Then, what can you say about the bijection μ?

(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 G act transitively on Ω and assume G is finite. Define an action of G...

Let G act transitively on Ω and assume G is finite. Define an action of G on the set Ω × Ω by putting (α,β) · g = (α · g, β · g). Let α ∈ Ω Show that G has the same number of orbits on Ω × Ω That Gα has on Ω

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...

The question is: Let G be a finite group, H, K be normal subgroups of G,...

The question is: Let G be a finite group, H, K be normal subgroups of G, and H∩K is also a normal subgroup of G. Using Homomorphism theorem ( or First Isomorphism theorem) prove that G/(H∩K) is isomorphism to a subgroup of (G/H)×(G/K). And give a example of group G with normal subgroups H and K such that G/(H∩K) ≆ (G/H)×(G/K), with explanation. I was trying to find some solutions for the isomorphism proof part, but they all seems to...

Let X be the set of all subsets of R whose complement is a finite set...

Let X be the set of all subsets of R whose complement is a finite set in R: X = {O ⊂ R | R − O is finite} ∪ {∅} a) Show that T is a topological structure no R. b) Prove that (R, X) is connected. c) Prove that (R, X) is compact.

Subjects