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  1. $(\Bbb Q/\Bbb Z,+)$ is an infinite abelian group in which every element has finite order.
  2. There exist an infinite group in which every element is of finite order.
  3. There do not exist any infinte group of which every element is of finite order and self inverse.
  4. Every infinite cyclic group is isomorphic to the additive group of all even integers.
  5. There exist a finite group which has infinte number of subgroups.

For 4th, taking $(\Bbb Z,+) \to (2\Bbb Z,+)$ won't be onto. So this should not be isomorphic. But it is. How? For the other parts, someone please help.

Akiva Weinberger
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Parul
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2 Answers2

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  1. See this duplicate

  2. See this duplicate.

  3. See this duplicate.

  4. See this duplicate.

  5. See this duplicate - a finite group has only finitely many different subgroups.

Community
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Dietrich Burde
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$1)$ true as $q(\mathbb Z + p/q)=\mathbb Z$

$2)$ true as $\mathbb Z/\mathbb Q$ works.

$3)$ true, consider $\mathbb Z_2^ \mathbb N$

$4)$ true, send $2k$ to $kg$, where $g$ is a generator for the cyclic group. Prove this is an isomorphism.

$5)$ False, subgroups are in particular subsets, so there can be at most $2^{|G|}$ subgroups of $G$.

Asinomás
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