Question

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.

Answer #1

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 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 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
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 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 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. 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 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 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 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.

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