Question

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

Answer #1

No.

Start with a lemma.

* Lemma:*
Let G be a group. If a ∈ G has infinite order, the a

* Proof:*
Let G be a group and let a ∈ G have infinite order. By way of
contradiction, assume a

The problem now follows. If a group has an element of infinite order, by the lemma it will have an infinite number of distinct elements (one for each power of a), therefore will be an infinite group.

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.

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.

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

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.

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.

Prove that a subset of a countably infinite set is finite or
countably infinite.

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.

3. Prove or disprove the following statement: If A and B are
finite sets, then |A ∪ B| = |A| + |B|.

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.

Prove or disprove: The group Q∗ is cyclic.

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