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

Expert Solution

Proof:

Elements that are linked by an automorphism have the same order, so the hypothesis in the theorem implies all non-identity elements of G have the same order.

Let p be a prime factor of |G|. Cauchy’s theorem gives us an element with order p, so all non-identity elements have order p. Thus |G| is a power of p. Since G is a non-trivial p-group, it has a non-trivial center. Elements that are linked by an automorphism are both in or both not in the center, so by the hypothesis of the theorem every non-identity element of G is in the center.

Thus G is abelian. Since each non-zero element has order p, we can view G as a vector space over Z/(p), necessarily finite-dimensional since G is finite. Picking a basis shows G ∼= (Z/(p))n for some n.

Elements that are linked by an automorphism have the same order, so the hypothesis in the theorem implies all non-identity elements of G have the same order.

Let p be a prime factor of |G|. Cauchy’s theorem gives us an element with order p, so all non-identity elements have order p. Thus |G| is a power of p.

Since G is a non-trivial p-group, it has a non-trivial center. Elements that are linked by an automorphism are both in or both not in the center, so by the hypothesis of the theorem every non-identity element of G is in the center.

Thus G is abelian. Since each non-zero element has order p, we can view G as a vector space over Z/(p), necessarily finite-dimensional since G is finite. Picking a basis shows G ∼= (Z/(p))n for some n.

Nipun Maity answered 2 years ago

(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 E be a finite extension of a ﬁeld F of prime characteristic p, and let...

let E be a finite extension of a ﬁeld F of prime characteristic p, and let K = F(Ep) be the subﬁeld of E obtained from F by adjoining the pth powers of all elements of E. Show that F(Ep) consists of all ﬁnite linear combinations of elements in Ep with coeﬃcients in F.

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...

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.

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.

let R = Z x Z. P be the prime ideal {0} x Z and S...

let R = Z x Z. P be the prime ideal {0} x Z and S = R - P. Prove that S^-1R is isomorphic to Q.

Let G be a group of order p am where p is a prime not dividing...

Let G be a group of order p am where p is a prime not dividing m. Show the following 1. Sylow p-subgroups of G exist; i.e. Sylp(G) 6= ∅. 2. If P ∈ Sylp(G) and Q is any p-subgroup of G, then there exists g ∈ G such that Q 6 gP g−1 ; i.e. Q is contained in some conjugate of P. In particular, any two Sylow p- subgroups of G are conjugate in G. 3. np ≡...

Consider n-dimensional finite affine space: F(n,p) over the field with prime p-elements. a. Show tha if...

Consider n-dimensional finite affine space: F(n,p) over the field with prime p-elements. a. Show tha if l and l' are two lines in F(n,p) containing the origin 0, then either l intersection l' ={0} or l=l' b. how many points lie on each line through the origin in F(n,p)? c. derive a formula for L(n,p), the number of lines through the origin in F(n,p)

11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]p, [2]p, ....

11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]p, [2]p, . . . , [p-1]p}. Prove that for y ≠ 0, Ly restricts to a bijective map Ly|s : S → S. 11.5 Prove Fermat's Little Theorem

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...

Question