Question

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

Answer #1

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 of even order. Show that the equation
x^2=e has even number of solutions in G

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.

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 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...

Let N be a normal subgroup of G. Show that the order 2 element
in N is in the center of G if N and Z_4 are isomorphic.

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.

2. Let a and b be elements of a group, G, whose identity element
is denoted by e. Prove that ab and ba have the same order. Show all
steps of proof.

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 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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