Question

Another way to show that in a free group, any non identity element is of infinite order

Answer #1

Show that in a free group, any nonidentity element is
of infinite order

Use the Mapping Property of free groups to show that
any non identity of a free group is of infinite ordet

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.

Give the definition of group. Show that the identity (or
neutral) element of G is
unique. Moreover, show that the inverse of an element g 2 G is
unique as well.

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

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.

True or False:
An infinite group must have an element of infinite order.
I know that the answer is false. Please give a detailed
explanation. Will gives thumbs up ASAP.

1. Let a and b be elements of a group, G, whose identity element
is denoted by e. Assume that a has order 7 and that a^(3)*b =
b*a^(3). Prove that a*b = b*a. Show all steps of proof.

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...

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.

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