3.2: Alternating Groups
- Page ID
- 131874
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Alternating groups \(A_n\) is the set of all even permutations associated with composition. \(|A_n|=\frac{n!}{2}\). \(A_n\) is a subgroup of the symmetric group \(S_n.\)
\( A_3=\{1, (1, 2, 3), (3, 2, 1)\}. A_3\) is a cyclic group of order \(3.\)
\(A_n\) is non abelian for \(n \ge 4\).
Proof:
Let \(a,b \in A_4\).
We will show that \(ab \ne ba, \; \forall a,b \in A_n\), \(n \ge 4\).
Note \(A_4=\{e, (12)(34), (13)(24), (14)(23), (123), (124), (134), (234), (432), (431), (421), (321) \}\).
Let \(a=(1,2)(3,4)\) and \(b=(4,1)(4,2)\).
Consider \(ab=(1,2)(3,4)(4,1)(4,2)=(1,3,4,2)\).
Further consider, \(ba=(4,1)(4,2)(1,2)(3,4)=(1,4,3,2)\).
Since \((1,3,4,2) \ne (1,4,3,2), \; \forall a,b \in A_4\), thus \(A_4\) is non-abelian.
Since a copy of \(A_4 \) is a subgroup of \(A_n, \; \forall n \ge 4\), \(A_n\) is non-abelian for \(N\ge 4\).◻
Find a cyclic subgroup of \(A_8\) that has order 4.
Consider the subgroup \(\{e, (1,2,3,4),(1,3)(2,4),(4,3,2,1)\}\) which is a subgroup of \(A_8\) of order 4.
Then consider \((1,2,3,4)(1,2,3,4)=(1,3)(2,4) \in A_8\).
Further \((1,2,3,4)(1,3)(2,4)=(1,4,3,2) \in A_8\).
Further \((1,2,3,4)(1,4,3,2)=e\).
Thus \(\{e, (1,2,3,4),(1,3)(2,4),(4,3,2,1)\}\), a subgroup of \(A_8\) that is order 4 and cyclic.
Find a non-cyclic subgroup of \(A_8\) that has order 4.
Consider \(\{e,(1,2),(2,3),(3,4)\}\) which is a subgroup of \(A_8\) of order 4.
Then consider \((1,2)^n\).
There is no \(n\) for which \((1,2)^n=(3,4) \text{ or } (2,3)\).
Similarly, there is no \(n\) for which \((2,3)^n=(1,2)\text{ or } (3,4)\).
Similarly, there is no \(n\) for which \((3,4)^n=(1,2)\text{ or } (2,3)\).
Thus \(\{e,(1,2),(2,3),(3,4)\}\) is not cyclic.
Version 2:
Let \(G=\{e,i,j,k\}\).
Let \(i=(1,2)(3,4)\), \(j=(5,6)(7,8)\) and \(k=(1,2)(3,4)(5,6)(7,8)\).
Consider \(ij=k\), \(ik=j\), and \(jk=i\).
Thus there is closure in \(G\).
Further, consider \(i^{-1}=i\), \(j^{-1}=j\), and \(k^{-1}=k\).
Thus, inverses exist for all elements of \(G\).
Note there is no \(n\) for which \(i^n=j\) or \(k\).
Similarly, there is no \(n\) for which \(j^n=i\) or \(k\).
Similarly, there is no \(n\) for which \(k^n=j\) or \(k\).
Since there is no \(<g> \in G\), \(G\) is not cyclic.
Find the conjugacy classes of \(A_3\).
Solution
\(\{e\}, \{(1, 2, 3)\}, \{(3, 2, 1)\}.\)
\([e]=\{\sigma^{-1} e\sigma \mid \sigma \in A_4\}=\{e\}.\)
\([(1, 2, 3)]=\{\sigma^{-1} (1, 2, 3) \sigma \mid \sigma \in A_4\}= \{e^{-1} (1, 2, 3) e, (1, 2, 3)^{-1} (1, 2, 3) (1, 2, 3), (3, 2, 1)^{-1} (1, 2, 3) (3, 2, 1)\}= \{(1, 2, 3)\}.\)
\([(3, 2, 1)]=\{\sigma^{-1} (3, 2, 1) \sigma \mid \sigma \in A_4\}= \{e^{-1} (3, 2, 1) e, (1, 2, 3)^{-1} (3, 2, 1) (1, 2, 3), (3, 2, 1)^{-1} (3, 2, 1) (3, 2, 1)\}= \{(3, 2, 1) \}.\)