Anki Deck Changes

Note 1: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: hx=y:u%$sF

modified

Before

Front

By what can we reduce the exponent of an element in a finite order Group?

Back

By what can we reduce the exponent of an element in a finite order Group?

In a group $G$ of finite order, for $a \in G$: \[a^m = a^{R_{\text{ord}(a)}(m)}\]This holds because:

$a^m = a^{m + \text{ord}(a)}$

$ = a^m \cdot a^{\text{ord}(a)}$

$ = a^m \cdot e = a^m$

After

Front

By what can we reduce the exponent of an element in a finite order Group?

Back

By what can we reduce the exponent of an element in a finite order Group?

In a group $G$ of finite order, for $a \in G$: \[a^m = a^{R_{\text{ord}(a)}(m)}\]This holds because:

$a^m = a^{m + \text{ord}(a)}$

$ = a^m \cdot a^{\text{ord}(a)}$

$ = a^m \cdot e = a^m$

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 Cloze
GUID: sxW-Trt$`+

deleted

Deleted Note

Front

$\mathbb{Z}_m^*$ is defined as?

Back

$\mathbb{Z}_m^*$ is defined as?

\[ \overset{\text{def}}{=} \ \{a \in \mathbb{Z}_m \ | \ \gcd(a, m) = 1\} \]

This is the set of all elements in $\mathbb{Z}_m$ that are coprime to $m$.

Current

Note has been deleted

Field-by-field Comparison

Field	Before	After
Text	~~<p>$\mathbb{Z}_m^*$ is defined as?</p>~~
Extra	~~<p>\[ \overset{\text{def}}{=} \ \{a \in \mathbb{Z}_m \ \| \ \gcd(a, m) = 1\} \]</p> <p>This is the set of all elements in $\mathbb{Z}_m$ that are coprime to $m$.</p>~~

Tags: ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function

Note 3: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: G&Y|dtr7^k

deleted

Deleted Note

Front

If the prime factorization of $m$ is $m = \prod_{i=1}^r p_i^{e_i}$ then $\varphi(m)$ is?

Back

If the prime factorization of $m$ is $m = \prod_{i=1}^r p_i^{e_i}$ then $\varphi(m)$ is?

For a prime and :\[\varphi(m) = \prod_{i=1}^r (p_i - 1)p_i^{e_i - 1}\]

Current

Note has been deleted

Field-by-field Comparison

Field	Before	After
Text	~~<p>If the prime factorization of $m$ is $m = \prod_{i=1}^r p_i^{e_i}$ then $\varphi(m)$ is?</p>~~
Extra	~~<p>For a prime and  :\[\varphi(m) = \prod_{i=1}^r (p_i - 1)p_i^{e_i - 1}\]<br></p>~~

Tags: ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::8._The_Group_Zₘ*_and_Euler's_Function

Field	Before	After
Text	~~<p>\(\mathbb{Z}_m^*\) is defined as?</p>~~
Extra	~~<p>\[ \overset{\text{def}}{=} \ \{a \in \mathbb{Z}_m \ \| \ \gcd(a, m) = 1\} \]</p> <p>This is the set of all elements in \(\mathbb{Z}_m\) that are coprime to \(m\).</p>~~

Field	Before	After
Text	~~<p>If the prime factorization of \(m\) is \(m = \prod_{i=1}^r p_i^{e_i}\) then \(\varphi(m)\) is?</p>~~
Extra	~~<p>For a prime and  :\[\varphi(m) = \prod_{i=1}^r (p_i - 1)p_i^{e_i - 1}\]<br></p>~~