For what \(m\) is \(\mathbb{Z}^*_m\) cyclic? (Theorem 5.15)
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Note Type: Horvath Classic
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Before
Front
Back
For what \(m\) is \(\mathbb{Z}^*_m\) cyclic? (Theorem 5.15)
The group \(\mathbb{Z}^*_m\) is cyclic if and only if:
• \(m = 2\)
• \(m = 4\)
• \(m = p^e\) (where p is an odd prime and \(e ≥ 1\))
• \(m = 2p^e\) (where p is an odd prime and \(e ≥ 1\))
Example: Is \(\mathbb{Z}^*_{19}\) cyclic? What is a generator? Yes, \(\mathbb{Z}^*_{19}\) is cyclic (since \(19\) is an odd prime).
• \(m = 2\)
• \(m = 4\)
• \(m = p^e\) (where p is an odd prime and \(e ≥ 1\))
• \(m = 2p^e\) (where p is an odd prime and \(e ≥ 1\))
Example: Is \(\mathbb{Z}^*_{19}\) cyclic? What is a generator? Yes, \(\mathbb{Z}^*_{19}\) is cyclic (since \(19\) is an odd prime).
- 2 is a generator.
- Powers of 2: 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1
- Other generators: 3, 10, 13, 14, 15
Why it doesn't contradict that very group of prime order is cyclic, with every element except the neutral element being a generator: \(\{2\} \cup [\text{all odd primes}]\)
\(= [\text{all primes}]\)After
Front
For what \(m\) is \(\mathbb{Z}^*_m\) cyclic? (Theorem 5.15)
Back
For what \(m\) is \(\mathbb{Z}^*_m\) cyclic? (Theorem 5.15)
The group \(\mathbb{Z}^*_m\) is cyclic if and only if:
• \(m = 2\)
• \(m = 4\)
• \(m = p^e\) (where p is an odd prime and \(e ≥ 1\))
• \(m = 2p^e\) (where p is an odd prime and \(e ≥ 1\))
Example: Is \(\mathbb{Z}^*_{19}\) cyclic? What is a generator? Yes, \(\mathbb{Z}^*_{19}\) is cyclic (since \(19\) is an odd prime).
• \(m = 2\)
• \(m = 4\)
• \(m = p^e\) (where p is an odd prime and \(e ≥ 1\))
• \(m = 2p^e\) (where p is an odd prime and \(e ≥ 1\))
Example: Is \(\mathbb{Z}^*_{19}\) cyclic? What is a generator? Yes, \(\mathbb{Z}^*_{19}\) is cyclic (since \(19\) is an odd prime).
- 2 is a generator.
- Powers of 2: 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1
- Other generators: 3, 10, 13, 14, 15
Why it doesn't contradict that every group of prime order is cyclic, with every element except the neutral element being a generator: \(\{2\} \cup [\text{all odd primes}]\)
\(= [\text{all primes}]\)Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Back | The group \(\mathbb{Z}^*_m\) is cyclic if and only if:<br>• \(m = 2\)<br>• \(m = 4\)<br>• \(m = p^e\) (where p is an odd prime and \(e ≥ 1\))<br>• \(m = 2p^e\) (where p is an odd prime and \(e ≥ 1\)) <br><br><b>Example:</b> Is \(\mathbb{Z}^*_{19}\) cyclic? What is a generator? Yes, \(\mathbb{Z}^*_{19}\) is cyclic (since \(19\) is an odd prime). <br><ul><li>2 is a generator.</li><li>Powers of 2: 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1</li><li>Other generators: 3, 10, 13, 14, 15</li></ul><div><b>Why it doesn't contradict </b>that <span style="color: rgb(51, 51, 51);">very group of </span>prime order<span style="color: rgb(51, 51, 51);"> is cyclic, with </span>every element except the neutral element being a generator<font color="#333333">: </font>\(\{2\} \cup [\text{all odd primes}]\)</div>\(= [\text{all primes}]\)<div><br></div> | The group \(\mathbb{Z}^*_m\) is cyclic if and only if:<br>• \(m = 2\)<br>• \(m = 4\)<br>• \(m = p^e\) (where p is an odd prime and \(e ≥ 1\))<br>• \(m = 2p^e\) (where p is an odd prime and \(e ≥ 1\)) <br><br><b>Example:</b> Is \(\mathbb{Z}^*_{19}\) cyclic? What is a generator? Yes, \(\mathbb{Z}^*_{19}\) is cyclic (since \(19\) is an odd prime). <br><ul><li>2 is a generator.</li><li>Powers of 2: 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1</li><li>Other generators: 3, 10, 13, 14, 15</li></ul><div><b>Why it doesn't contradict </b>that <span style="color: rgb(51, 51, 51);">every group of </span>prime order<span style="color: rgb(51, 51, 51);"> is cyclic, with </span>every element except the neutral element being a generator<font color="#333333">: </font>\(\{2\} \cup [\text{all odd primes}]\)</div>\(= [\text{all primes}]\)<div><br></div> |