Question

In: Advanced Math

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)

Solutions

Expert Solution

this is different process by calculation.i have did by theory.but for you I am again uploading by calculating.please like this please.dont give dislikes.


Related Solutions

Let G be a group of order 40. Can G have an element of order 3?
Let G be a group of order 40. Can G have an element of order 3?
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 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 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.
abstract algebra Let G be a finite abelian group of order n Prove that if d...
abstract algebra Let G be a finite abelian group of order n Prove that if d is a positive divisor of n, then G has a subgroup of order d.
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?
(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...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT