Question

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.

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 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...

Suppose : phi :G -H is a group isomorphism . If N is a normal
subgroup of G then phi(N) is a normal subgroup of H. Prove it is a
subgroup and prove it is normal?

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...

4. Let f : G→H be a group homomorphism. Suppose a∈G is an
element of finite order n.
(a) Prove that f(a) has finite order k, where k is a divisor of
n.
(b) If f is an isomorphism, prove that k=n.

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. 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.

Let G be a group and suppose H = {g5 : g ∈ G} is a
subgroup of G.
(a) Prove that H is normal subgroup of G.
(b) Prove that every element in G/H has order at most 5.

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].

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 13 minutes ago

asked 29 minutes ago

asked 31 minutes ago

asked 32 minutes ago

asked 36 minutes ago

asked 44 minutes ago

asked 55 minutes ago

asked 58 minutes ago

asked 59 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago