Question

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 show that aH and bH contain the same number of elements in G.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let G be a finite group, and suppose that H is normal subgroup of G. Show...
Let G be a finite group, and suppose that H is normal subgroup of G. Show that, for every g ∈ G, the order of gH in G/H must divide the order of g in G. What is the order of the coset [4]42 + 〈[6]42〉 in Z42/〈[6]42〉? Find an example to show that the order of gH in G/H does not always determine the order of g in G. That is, find an example of a group G, and...
Show that if H is a subgroup of index 2 in a finite group G, then...
Show that if H is a subgroup of index 2 in a finite group G, then every left coset of H is also a right coset of H. *** I have the answer but I am really looking for a thorough explanation. Thanks!
Let G be a group, and H a subgroup of G, let a,b?G prove the statement...
Let G be a group, and H a subgroup of G, let a,b?G prove the statement or give a counterexample: If aH=bH, then Ha=Hb
Let G be a finite group and H be a subgroup of G. Prove that if...
Let G be a finite group and H be a subgroup of G. Prove that if H is only subgroup of G of size |H|, then H is normal in G.
Let G be a finite group and let H be a subgroup of order n. Suppose...
Let G be a finite group and let H be a subgroup of order n. Suppose that H is the only subgroup of order n. Show that H is normal in G. Hint: Consider the subgroup aHa-1 of G. Please explain in detail!
Searches related to Let H be a subgroup of G. Define an action of H on...
Searches related to Let H be a subgroup of G. Define an action of H on G by h*g=gh^-1. Prove this is a group action. Why do we use h^-1? Prove that the orbit of a in G is the coset aH.
a) Let H be a subgroup of a group G satisfying [G ∶ H] = 2....
a) Let H be a subgroup of a group G satisfying [G ∶ H] = 2. If there are elements a, b ∈ G such that ab ∈/ H, then prove that either a ∈ H or b ∈ H. (b) List the left and right cosets of H = {(1), (23)} in S3. Are they the same collection?
Prove the following theorem: Let φ: G→G′ be a group homomorphism, and let H=ker(φ). Let a∈G.Then...
Prove the following theorem: Let φ: G→G′ be a group homomorphism, and let H=ker(φ). Let a∈G.Then the set (φ)^{-1}[{φ(a)}] ={x∈G|φ(x)} =φ(a) is the left coset aH of H, and is also the right coset Ha of H. Consequently, the two partitions of G into left cosets and into right cosets of H are the same
Let G be a finite group and let P be a Sylow p-subgroup of G. Suppose...
Let G be a finite group and let P be a Sylow p-subgroup of G. Suppose H is a normal subgroup of G. Prove that HP/H is a Sylow p-subgroup of G/H and that H ∩ P is a Sylow p-subgroup of H. Hint: Use the Second Isomorphism theorem.
Let H be a subgroup of a group G. Let ∼H and ρH be the equivalence...
Let H be a subgroup of a group G. Let ∼H and ρH be the equivalence relation in G introduced in class given by x∼H y⇐⇒x−1y∈H, xρHy⇐⇒xy−1 ∈H. The equivalence classes are the left and the right cosets of H in G, respectively. Prove that the functionφ: G/∼H →G/ρH given by φ(xH) = Hx−1 is well-defined and bijective. This proves that the number of left and right cosets are equal.