Question

let G be a finite group of even order. Show that the equation x^2=e has even number of solutions in G

Answer #1

For an abelian group G, let tG = {x E G: x has finite order}
denote its torsion subgroup.
Show that t defines a functor Ab -> Ab if one defines t(f) =
f|tG (f restricted on tG) for every homomorphism f.
If f is injective, then t(f) is injective.
Give an example of a surjective homomorphism f for which t(f)
is not surjective.

Let G be a group. g be an element of G. if
<g^2>=<g^4> show that order of g is finite.

: (a) Let p be a prime, and let G be a finite Abelian group.
Show that Gp = {x ∈ G | |x| is a power of p} is a subgroup of G.
(For the identity, remember that 1 = p 0 is a power of p.) (b) Let
p1, . . . , pn be pair-wise distinct primes, and let G be an
Abelian group. Show that Gp1 , . . . , Gpn form direct sum in...

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!

Prove that if (G, ·) is a finite group of even order, then
there always exists an element g∈G such that g ≠ 1 and
g2=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...

(A) Show that if a2=e for all elements a in a group
G, then G must be abelian.
(B) Show that if G is a finite group of even order, then there
is an a∈G such that a is not the identity and a2=e.
(C) Find all the subgroups of Z3×Z3. Use
this information to show that Z3×Z3 is not
the same group as Z9.
(Abstract Algebra)

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 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 of order p^2, where p is a prime.
Show that G must have a subgroup of order p.
please show with notation if possible

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