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?

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let G be a group acting on a set S, and let H be a group...

Let G be a group acting on a set S, and let H be a group acting on a set T. The product group G × H acts on the disjoint union S ∪ T as follows. For all g ∈ G, h ∈ H, s ∈ S and t ∈ T, (g, h) · s = g · s, (g, h) · t = h · t. (a) Consider the groups G = C4, H = C5, each acting...

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 S be a non-empty set (finite or otherwise) and Σ the group of permutations on...

Let S be a non-empty set (finite or otherwise) and Σ the group of permutations on S. Suppose ∼ is an equivalence relation on S. Prove (a) {ρ ∈ Σ : x ∼ ρ(x) (∀x ∈ S)} is a subgroup of Σ. (b) The elements ρ ∈ Σ for which, for every x and y in S, ρ(x) ∼ ρ(y) if and only if x ∼ y is a subgroup of Σ.

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)

(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, +)

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 G be a group. The center of G is the set Z(G) = {g∈G |gh...

Let G be a group. The center of G is the set Z(G) = {g∈G |gh = hg ∀h∈G}. For a∈G, the centralizer of a is the set C(a) ={g∈G |ga =ag } (a)Prove that Z(G) is an abelian subgroup of G. (b)Compute the center of D4. (c)Compute the center of the group G of the shuffles of three objects x1,x2,x3. ○n: no shuffling occurred ○s12: swap the first and second items ○s13: swap the first and third items ○s23:...

Let V be a finite dimensional vector space over R. If S is a set of...

Let V be a finite dimensional vector space over R. If S is a set of elements in V such that Span(S) = V , what is the relationship between S and the basis of V ?

Let G be a group. Consider the set G with a new operation ∗ given by...

Let G be a group. Consider the set G with a new operation ∗ given by a ∗ b = ba. Show that (G, ∗) is a group isomorphic to the original group G. Give an explicit isomorphism.

Subjects