4.5: Isormormophism theorems
- Page ID
- 132501
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)An isomorphism between two groups is a bijective mapping between two groups that preserves the group structure.
Let \(G\) and \(H\) be groups. Let \( \phi: G \rightarrow H\).
An isomorphism from \(G\) to \(H\) is a function \(\phi: G \rightarrow H\) that satisfies the following two properties:
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Homomorphism property: For any \(g_1, g_2 \in G\), the operation in \(H\) corresponding to \(\phi(g_1)\) and \(\phi(g_2)\) is the same as the operation in \(G\) corresponding to \(g_1\) and \(g_2\). In other words, \(\phi(g_1g_2) = \phi(g_1) \cdot \phi(g_2)\), where \(\cdot\) represents the group operation.
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Bijective property: The function \(\phi\) is one-to-one (injective) and onto (surjective), meaning that every element in \(H\) is the image of exactly one element in \(G\) under \(\phi\).
If an isomorphism exists between two groups \(G\) and \(H\), we say that \(G\) and \(H\) are isomorphic, denoted by \(G \cong H\). This means the two groups have the same group structure, even though their elements and operation symbols may differ.
Isomorphisms are significant in mathematics because they establish a correspondence between different groups, allowing us to understand the structure and properties of one group by studying the properties of another group.
Recall: The kernel of \(\phi\), denoted by \(\text{Ker}(\phi)\), is defined as the set of elements in \(G\) that map to the identity element of \(H\). the image of \(\phi\) is the set of all values that \(\phi\) takes on as it maps elements from \(G\) to \(H\), denoted by \(\text{Im}(\phi) == \{\phi(g) ,|, g \in G\}\).
Let \(G\) and \(H\) be groups. Let \( \phi: G \rightarrow H\) be a group homomorphism. Then, the factor group \(G/\text{Ker}(\phi)\) is isomorphic to the image of \(G,\) under \(\phi\), \(\text{Im}(\phi)\). That is, \(G/\text{Ker}(\phi) \cong \text{Im}(\phi)\)
Consider the groups \(G = (\mathbb{Z}, +)\) (the group of integers under addition) and \(H = (\mathbb{Z}/4\mathbb{Z}, +4)\) (the group of integers modulo 4 under addition modulo 4). Define the homomorphism\(\varphi: \mathbb{Z} \rightarrow \mathbb{Z}/4\mathbb{Z}\) as\(\varphi(x) = x \mod 4\).
The kernel of\(\varphi\) is\(\text{Ker}(\varphi) = {x \in \mathbb{Z} ,|, \varphi(x) = 0} = {x \in \mathbb{Z} ,|, x \mod 4 = 0} = 4\mathbb{Z}\) (the subgroup of multiples of 4).
By applying the First Isomorphism Theorem, we have\(G/\text{Ker}(\varphi) \cong \text{Im}(\varphi)\). In this case,\(G/\text{Ker}(\varphi)\) is isomorphic to the image of\(G\) under\(\varphi\).
Using the theorem, we find that\(G/\text{Ker}(\varphi) \cong H\), which means\((\mathbb{Z}/4\mathbb{Z}, +4)\) is isomorphic to itself.
Let \(G\) be a group,\(H\) and\(K\) be subgroups of \(G\) such that \(H \leq G\). Then, the set \(HK = {hk ,|, h \in H, k \in K}\) is a subgroup of\(G\), and\(H \cap K\) is a normal subgroup of\(H\). Moreover, the factor group\((HK)/(H \cap K)\) is isomorphic to\(K/(H \cap K)\).
That is, \((HK)/(H \cap K) \cong K/(H \cap K)\)
Consider the group \(G = (\mathbb{Z}, +)\) (the group of integers under addition). Let \(H = (2\mathbb{Z}, +)\) be the subgroup of even integers, and \(K = (3\mathbb{Z}, +)\) be the subgroup of multiples of 3.
In this case, \(H\) and \(K\) are both normal subgroups of \(G\).
By applying the Second Isomorphism Theorem, we have \((HK)/(H \cap K) \cong K/(H \cap K)\). Here, \((HK)/(H \cap K)\) is isomorphic to \(K/(H \cap K)\).
Using the theorem, we find that \((2\mathbb{Z} + 3\mathbb{Z}) / (2\mathbb{Z} \cap 3\mathbb{Z}) \cong (3\mathbb{Z}) / (2\mathbb{Z} \cap 3\mathbb{Z)}\).
In this example, \((2\mathbb{Z} + 3\mathbb{Z}) / (2\mathbb{Z} \cap 3\mathbb{Z})\) represents the set of all integers that are divisible by both 2 and 3, which is equivalent to \((6\mathbb{Z}) / (6\mathbb{Z}) = {0}\) (the trivial group). Similarly, \((3\mathbb{Z}) / (2\mathbb{Z} \cap 3\mathbb{Z})\) represents the set of all integers that are divisible by 3 but not divisible by 2, which is isomorphic to \((\mathbb{Z}/3\mathbb{Z}, +3)\) (the group of integers modulo 3 under addition modulo 3).
Therefore, we conclude that \({0} \cong \mathbb{Z}/3\mathbb{Z}\).
Let \(G\) be a group, and let \(N\) and \(M\) be normal subgroups of \(G\), with \(N \leq M\). Then, the factor group \(M/N\) is isomorphic to a subgroup of \(G/N\).
That is, \(M/N \cong (M/N) / (N/N)\)
Consider the group \(G = (\mathbb{Z}, +)\) (the group of integers under addition). Let \(N = (6\mathbb{Z}, +)\) be the normal subgroup of multiples of 6, and \(M = (9\mathbb{Z}, +)\) be the normal subgroup of multiples of 9.
In this case, \(N \leq M\).
By applying the Third Isomorphism Theorem, we have \(M/N \cong (M/N) / (N/N)\).
Using the theorem, we find that \((9\mathbb{Z}) / (6\mathbb{Z}) \cong (9\mathbb{Z} / 6\mathbb{Z}) / (6\mathbb{Z} / 6\mathbb{Z})\).
The factor group \((9\mathbb{Z}) / (6\mathbb{Z})\) represents the set of all integers that are multiples of 9, modulo 6. This is isomorphic to \((\mathbb{Z}/6\mathbb{Z}, +6)\) (the group of integers modulo 6 under addition modulo 6).
Similarly, \((9\mathbb{Z} / 6\mathbb{Z}) / (6\mathbb{Z} / 6\mathbb{Z})\) represents the set of cosets of \((6\mathbb{Z} / 6\mathbb{Z})\) within \((9\mathbb{Z} / 6\mathbb{Z})\). Since \((6\mathbb{Z} / 6\mathbb{Z})\) is the trivial group, there is only one coset, which is \({6\mathbb{Z}}\).
Therefore, we conclude that \(\mathbb{Z}/6\mathbb{Z} \cong {6\mathbb{Z}}\) (the trivial group).