Question

suppose every element of a group G has order dividing 2. Show that G is an abelian group.

There is another question on this, but I can't understand the writing at all...

Answer #1

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

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.

Suppose that G is abelian group of order 16, and in computing the
orders of its elements, you come across an element of order 8 and 2
elements of order 2. Explain why no further computations are needed
to determine the isomorphism class of G. provide explanation
please.

Let G be a non-abelian group of order p^3 with p prime.
(a) Show that |Z(G)| = p. (b) Suppose a /∈ Z(G). Show that
|NG(a)| = p^2 .
(c) Show that G has exactly p 2 +p−1 conjugacy classes (don’t
forget to count the classes of the elements of Z(G)).

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

Suppose that G is a finite Abelian group that has exactly one
subroup for each divisor of the order of G. Show that G is cyclic.
provide explanation.

1(a) Suppose G is a group with p + 1 elements of order p , where
p is prime. Prove that G is not cyclic.
(b) Suppose G is a group with order p, where p is prime. Prove
that the order of every non-identity element in G is p.

Let n be a positive integer. Show that every abelian group of
order n is cyclic if and only if n is not divisible by the square
of any prime.

find all generators of Z. let "a" be a group element that has
infinite order. Find all the generators of . Please prove and
explain in detail please use definions and theorems. please i
reallly want to understand this.

Let G be an abelian group and S ≤ G. Show that S ⊲ G and that
G/S is abelian
I need an explanation with some details

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