Question

(b) If H is a *p*-subgroup of a finite group G, prove
that H is contained in a Sylow *p*-subgroup of G.

[Hint: **Consider the H-conjugacy class equation for the
set of all Sylowp-subgroups of G.]**

Answer #1

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 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!

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 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...

(Abstract algebra) Let G be a group and let H and K be subgroups
of G so that H is not contained in K and K is not contained in H.
Prove that H ∪ K is not a subgroup of 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 group with subgroups H and K.
(a) Prove that H ∩ K must be a subgroup of G.
(b) Give an example to show that H ∪ K is not necessarily a
subgroup of G.
Note: Your answer to part (a) should be a general proof that the
set H ∩ K is closed under the operation of G, includes the identity
element of G, and contains the inverse in G of each of its
elements,...

Let G be a finite group and let H, K be normal subgroups of G.
If [G : H] = p and [G : K] = q where p and q are distinct primes,
prove that pq divides [G : H ∩ K].

Prove that if A is a subgroup of G and B is a subgroup of H,
then the direct product A × B is a subgroup of G × H.
Show all steps. Note that AXB is nonempty since the identity e
is a part of A X B. Remains only to show that A X B is closed under
multiplication and inverses.

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