Question

*Prove that if (G, ·) is a finite group of even order, then
there always exists an element g∈G such that g* ≠ 1 and
g^{2}=1.

Answer #1

Can there be an element of infinite order in a finite group?
Prove or disprove.

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 group (not necessarily an Abelian group) of order
425. Prove that G must have an element of order 5. Note, Sylow
Theorem is above us so we can't use it. We're up to Finite Orders.
Thank you.

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

Is it possible for a group G to contain a non-identity element
of finite order and also an element of infinite order? If yes,
illustrate with an example. If no, give a convincing explanation
for why it is not possible.

Let G be a group of order 4. Prove that either G is generated by
a single element or g^2 =1 for all g∈G.

LetG be a group (not necessarily an Abelian group) of order 425.
Prove that G must have an element of order 5.

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

Prove that if F is a field and K = FG for a finite
group G of automorphisms of F, then there are only finitely many
subfields between F and K.
Please help!

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