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.

Expert Solution

IF YOU HAVE ANY DOUBTS COMMENT BELOW I WILL BE TTHERE TO HELP YOU..ALL THE BEST..

AS FOR GIVEN DATA..

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

EXPLANATION ::-

a. associate degree automorphism maps the middle of G to the center of G. Thus, below your condition the middle of G is that the full cluster and it's so abelian, or the middle is trivial.

b. Note that associate degree automorphism preserves the order of a part. so the transitivity implies that each one component of G except one have the identical order n. And this n (thus) needs to be prime. particularly, the middle is non-trivial.

Thus, G may be a finite abelian p-group during which all components (except 1) have order n and n is prime. That is, it's associate degree elementary abelian p-group.

I HOPE YOU UNDERSTAND..

PLS RATE THUMBS UP..ITS HELPS ME ALOT..

THANK YOU...!!

Colby Messinger answered 2 years ago

Let G be finite with |G| > 1. If Aut(G) acts transitively on G − {e} then G ∼= (Z/(p))n for some prime p.

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 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...

(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, +)

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...

show that if H is a p sylow subgroup of a finite group G then for...

show that if H is a p sylow subgroup of a finite group G then for an arbitrary x in G x^-1 H x is also a p sylow subgroup of G

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 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?

Let G be a finite group and p be a prime number.Suppose that pr divides the...

Let G be a finite group and p be a prime number.Suppose that pr divides the order of G. Show that G has a proper subgroup of order pr.

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...

Question