# a

1. ### |<a>| = |a|

how to prove this: Let G be a group and a e G an element of finite order. Then |<a>| = |a|. Thank you!! ;]
2. ### Corollary 1. |a| divides |G|

I just would like to ask for the proof of this Corollary: |a| divides |G|. In a finite group, the order of each element of the group divides the order of the group. (The theorem is Lagrange's Theorem: |H| divides |G|. If G is a finite group and H is a subgroup of G, then |H| divides |G|...
3. ### Proving |a+b| = |a|+|b|

Hi guys, I'm doing a case by case proof of |a+b| \leq |a|+|b| from Spivak, and he says: When a \geq 0 and b \leq 0, we must prove that: |a+b| \leq a-b I'm a bit stuck on his line of reasoning, could someone explain why we have to prove the above?
4. ### Prove that if |A| ≤ |B| and |B| ≤ |C| then |A| ≤ |C| (Cardinality)

Hello, Can someone do this question please ? I can't do it. Prove that if |A| ≤ |B| and |B| ≤ |C| then |A| ≤ |C| Note: |A| ≤ |B| means "there exist an injection from set A into set B" and so on for the rest. I would really appreciate it if someone can help me because I find this...
5. ### Prove that if |A| ≤ |B| and |B| ≤ |C| then |A| ≤ |C|

Hello, Can anyone help me on this question please ? Prove that if |A| ≤ |B| and |B| ≤ |C| then |A| ≤ |C|