Anki Deck Changes

Note 1: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: qAyHDnFN7L

added

New Note

Front

What is the number of generators of $\mathbb{Z}_n^*$?

Back

What is the number of generators of $\mathbb{Z}_n^*$?

1. verify that $\mathbb{Z}_n^*$is cyclic (iff n = 2, 4, $p^e$, $2p^e$, with $e \ge 1$ and $p$ is an odd prime)
2. if $\mathbb{Z}_n^*$ is cyclic then it is isomorphic to $\mathbb{Z}_{\varphi(n)}^+$ (by lemma)
3. the number of generators of $\mathbb{Z}_{\varphi(n)}^+$ is $\varphi(\varphi(n))$ as it is the number of coprime elements of the group

Field-by-field Comparison

Field	Before	After
Front		What is the number of generators of $\mathbb{Z}_n^*$?
Back		1. verify that $\mathbb{Z}_n^$is cyclic (iff n = 2, 4, $p^e$, $2p^e$, with $e \ge 1$ and $p$ is an odd prime)<br>2. if $\mathbb{Z}_n^$ is cyclic then it is isomorphic to $\mathbb{Z}_{\varphi(n)}^+$ (by lemma) <br>3. the number of generators of $\mathbb{Z}_{\varphi(n)}^+$ is $\varphi(\varphi(n))$ as it is the number of coprime elements of the group

Tags: ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups

Note 2: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: pD<6]f{)8D

added

New Note

Front

What is the number of generators of $\mathbb{Z}_{25}^* $?

Back

What is the number of generators of $\mathbb{Z}_{25}^* $?

it is $\varphi(\varphi(25)) = |\mathbb{Z}_{\varphi(25)}^*| = |\mathbb{Z}_{20}^*| = 8$ ( 1, 3, 7, 9, 11, 13, 17, 19 )

Field-by-field Comparison

Field	Before	After
Front		What is the number of generators of $\mathbb{Z}_{25}^* $?
Back		it is $\varphi(\varphi(25)) = \|\mathbb{Z}_{\varphi(25)}^\| = \|\mathbb{Z}_{20}^\| = 8$ ( 1, 3, 7, 9, 11, 13, 17, 19 )

Tags: ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups

Note 3: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: M035/^ZEJ$

added

New Note

Front

What is the number of subgroups of $\mathbb{Z}_n$?

Back

What is the number of subgroups of $\mathbb{Z}_n$?

it is the number of divisors of $n$
if $n$ is written $n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k}$ then it is $\prod_{i=1}^k (e_i+1)$

Field-by-field Comparison

Field	Before	After
Front		What is the number of subgroups of $\mathbb{Z}_n$?
Back		it is the number of divisors of $n$<br>if $n$ is written $n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k}$ then it is $\prod_{i=1}^k (e_i+1)$

Tags: ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups

Anki Deck Changes

Note 1: ETH::DiskMat

Previous

New Note

Front

Back

Note 2: ETH::DiskMat

Previous

New Note

Front

Back

Note 3: ETH::DiskMat

Previous

New Note

Front

Back

Field	Before	After
Front		What is the number of generators of \(\mathbb{Z}_n^*\)?
Back		1. verify that \(\mathbb{Z}_n^\)is cyclic (iff n = 2, 4, \(p^e\), \(2p^e\), with \(e \ge 1\) and \(p\) is an odd prime)<br>2. if \(\mathbb{Z}_n^\) is cyclic then it is isomorphic to \(\mathbb{Z}_{\varphi(n)}^+\) (by lemma) <br>3. the number of generators of \(\mathbb{Z}_{\varphi(n)}^+\) is \(\varphi(\varphi(n))\) as it is the number of coprime elements of the group

Field	Before	After
Front		What is the number of subgroups of \(\mathbb{Z}_n\)?
Back		it is the number of divisors of \(n\)<br>if \(n\) is written \(n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k}\) then it is \(\prod_{i=1}^k (e_i+1)\)