Question

Let G be a non-trivial finite group, and let H < G be a proper subgroup. Let X be the set of conjugates of H, that is, X = {aHa^(−1) : a ∈ G}. Let G act on X by conjugation, i.e., g · (aHa^(−1) ) = (ga)H(ga)^(−1) .

Prove that this action of G on X is transitive.

Use the previous result to prove that G is not covered by the conjugates of H, i.e., G does not equal S a∈G aHa^(−1) .

Answer #1

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!

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 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.

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 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...

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.

Let
G be a finite group. There are 2 ways of getting a subgroup of G,
which are {e} and G. Now, prove the following : If |G|>1 is not
prime, then G has a subgroup other than the 2 groups which are
mentioned in the above.

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