Question

In: Advanced Math

A) Suppose a group G has order 35 and acts on a set S consisting of...

A) Suppose a group G has order 35 and acts on a set S consisting of four elements. What can you say about the action?

B) What happens if |G|=28? |G|=30?

Expert Solution

A) This action induces a group homomorphism of into the group of permutations of , given by for all and . Note that , so that . If then by first isomorphism theorem, we know that

By Lagrange's theorem, we get , so that divides . Again, since is a subgroup, by Lagrange, divides . Therefore, divides , so that divides . Thus, , which implies . Thus, the action is trivial, given by for all and .

B)

Suppose that . The same argument as above shows that divides , so that divides . Thus, there are three possibilities:

If holds then the action is trivial.

Suppose that . The same argument as above shows that divides , so that divides . Thus, there are three possibilities:

If holds then the action is trivial.

Colby Messinger answered 1 year ago

Suppose that a group G acts on its power set P(G) by conjugation. a) If H≤G,...

Suppose that a group G acts on its power set P(G) by conjugation. a) If H≤G, prove that the normalizer of N(H) is the largest subgroup K of G such that HEK. In particular, if G is finite show that |H| divides |N(H)|. b) Show that H≤G⇐⇒every element B∈OH is a subgroup of G with B∼=H. c) Prove that HEG⇐⇒|OH|= 1.

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...

If G is a finite group and Aut(G) acts transitively on G^# = G − 1,...

If G is a finite group and Aut(G) acts transitively on G^# = G − 1, then prove that G is an elementary abelian group.

If G is a group, show that the set of automorphisms of G and the set...

If G is a group, show that the set of automorphisms of G and the set of inner automorphisms of G are both groups.

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?

An automorphism of a group G is an isomorphism from G to G. The set of...

An automorphism of a group G is an isomorphism from G to G. The set of all automorphisms of G forms a group Aut(G), where the group multiplication is the composition of automorphisms. The group Aut(G) is called the automorphism group of group G. (a) Show that Aut(Z) ≃ Z2. (Hint: consider generators of Z.) (b) Show that Aut(Z2 × Z2) ≃ S3. (c) Prove that if Aut(G) is cyclic then G is abelian.

Explanations are much appreciated. Let G be a group of order 105. Prove that G has...

Explanations are much appreciated. Let G be a group of order 105. Prove that G has a proper normal subgroup.

a symmetric group S5 acts on the set X5 = {(i, j) : i, j ∈...

a symmetric group S5 acts on the set X5 = {(i, j) : i, j ∈ {1, 2, 3, 4, 5}}. S5 will also act on this set. Consider the subgroup H = <(1, 2)(3, 4), (1, 3)(2, 4)>≤ S5. (a) Find the orbits of H in this action. Justify your answers. (b) For each orbit find the stabiliser one of its members. Justify your answers. action is this t.(i,j)=(t(i),t(j))

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:...

Suppose that a set G has a binary operation on it that has the following properties:...

Suppose that a set G has a binary operation on it that has the following properties: 1. The operation ◦ is associative, that is: for all a,b,c ∈ G, a◦(b◦c)=(a◦b)◦c 2. There is a right identity, e: For all a∈G a◦e=a 3. Every element has a right inverse: For all a∈G there is a^-1 such that a◦a^-1=e Prove that this operation makes G a group. You must show that the right inverse of each element is a left inverse and...