Let G be a group of order 40. Can G have an element of order 3?

Expert Solution

Colby Messinger answered 1 year ago

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 group and a be an element of G. Let φ:Z→G be a...

Let G be a group and a be an element of G. Let φ:Z→G be a map defined by φ(n) =a^{n} for all n∈Z. a)Show that φ is a group homomorphism. b) Find the image ofφ, i.e.φ(Z), and prove that it is a subgroup ofG.

Let G be a group, and let a ∈ G be a fixed element. Define a...

Let G be a group, and let a ∈ G be a fixed element. Define a function Φ : G → G by Φ(x) = ax−1a−1. Prove that Φ is an isomorphism is and only if the group G is abelian.

Let G be a group of order 42 = 2 * 3 * 7 (a) Let...

Let G be a group of order 42 = 2 * 3 * 7 (a) Let P7 be a Sylow 7-subgroup of G and let P3 be a Sylow 3-subgroup of G . What are the orders of P3 and P7? (b) Prove that P7 is the unique Sylow 7-subgroup of G and that P7 is normal. (c) Prove that P3P7 is a subgroup of G (d) Prove that P3P7 is a normal subgroup of G . (e) Let P2...

Let G be a cyclic group generated by an element a. a) Prove that if an...

Let G be a cyclic group generated by an element a. a) Prove that if an = e for some n ∈ Z, then G is finite. b) Prove that if G is an infinite cyclic group then it contains no nontrivial finite subgroups. (Hint: use part (a))

Abstract Algebra Let G be a group of order 351. Prove that G is not simple.

Let B be a finite commutative group without an element of order 2. Show the mapping...

Let B be a finite commutative group without an element of order 2. Show the mapping of b to b2 is an automorphism of B. However, if |B| = infinity, does it still need to be an automorphism?

Let G be a nonabelian group of order 253=23(11), let P<G be a Sylow 23-subgroup and...

Let G be a nonabelian group of order 253=23(11), let P<G be a Sylow 23-subgroup and Q<G a Sylow 11-subgroup. a. What are the orders of P and Q. (Explain and include any theorems used). b. How many distinct conjugates of P and Q are there in G? n23? n11? (Explain, include any theorems used). c. Prove that G is isomorphic to the semidirect product of P and Q.

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.

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 ≡...

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