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<h1>Конечные группы, Абелевы группы<a class="headerlink" href="#id1" title="Ссылка на этот заголовок">¶</a></h1>
<p>Sage поддерживает вычисления с группами перестановок, конечными
классическими группами (как, например, <img class="math" src="_images/math/ea408e1735df8926cb4055d216f8143b53224cdc.png" alt="SU(n,q)"/>), конечными
матричными группами (с собственными генераторами) и группами Абеля
(даже с бесконечными). Многое из этого осуществляется посредством
интерфейса к GAP.</p>
<p>Например, чтобы построить группу перестановок, надо задать список генераторов:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="go">sage: G = PermutationGroup(['(1,2,3)(4,5)', '(3,4)'])</span>
<span class="go">sage: G</span>
<span class="go">Permutation Group with generators [(3,4), (1,2,3)(4,5)]</span>
<span class="go">sage: G.order()</span>
<span class="go">120</span>
<span class="go">sage: G.is_abelian()</span>
<span class="go">False</span>
<span class="go">sage: G.derived_series() # random output</span>
<span class="go">[Permutation Group with generators [(1,2,3)(4,5), (3,4)],</span>
<span class="go"> Permutation Group with generators [(1,5)(3,4), (1,5)(2,4), (1,3,5)]]</span>
<span class="go">sage: G.center()</span>
<span class="go">Subgroup of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) generated by [()]</span>
<span class="go">sage: G.random_element() # random output</span>
<span class="go">(1,5,3)(2,4)</span>
<span class="go">sage: print latex(G)</span>
<span class="go">\langle (3,4), (1,2,3)(4,5) \rangle</span>
</pre></div>
</div>
<p>Также можно получить символьную таблицу в формате LaTeX:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="go">sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])</span>
<span class="go">sage: latex(G.character_table())</span>
<span class="go">\left(\begin{array}{rrrr}</span>
<span class="go">1 & 1 & 1 & 1 \\</span>
<span class="go">1 & 1 & -\zeta_{3} - 1 & \zeta_{3} \\</span>
<span class="go">1 & 1 & \zeta_{3} & -\zeta_{3} - 1 \\</span>
<span class="go">3 & -1 & 0 & 0</span>
<span class="go">\end{array}\right)</span>
</pre></div>
</div>
<p>Sage также включает в себя классические и матричные группы для конечных полей:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="go">sage: MS = MatrixSpace(GF(7), 2)</span>
<span class="go">sage: gens = [MS([[1,0],[-1,1]]),MS([[1,1],[0,1]])]</span>
<span class="go">sage: G = MatrixGroup(gens)</span>
<span class="go">sage: G.conjugacy_class_representatives()</span>
<span class="go"> [</span>
<span class="go"> [1 0]</span>
<span class="go"> [0 1],</span>
<span class="go"> [0 1]</span>
<span class="go"> [6 1],</span>
<span class="go"> ...</span>
<span class="go"> [6 0]</span>
<span class="go"> [0 6]</span>
<span class="go"> ]</span>
<span class="go">sage: G = Sp(4,GF(7))</span>
<span class="go">sage: G._gap_init_()</span>
<span class="go">'Sp(4, 7)'</span>
<span class="go">sage: G</span>
<span class="go">Symplectic Group of rank 2 over Finite Field of size 7</span>
<span class="go">sage: G.random_element() # random output</span>
<span class="go">[5 5 5 1]</span>
<span class="go">[0 2 6 3]</span>
<span class="go">[5 0 1 0]</span>
<span class="go">[4 6 3 4]</span>
<span class="go">sage: G.order()</span>
<span class="go">276595200</span>
</pre></div>
</div>
<p>Также можно производить вычисления с группами Абеля (конечными и бесконечными):</p>
<div class="highlight-python"><div class="highlight"><pre><span class="go">sage: F = AbelianGroup(5, [5,5,7,8,9], names='abcde')</span>
<span class="go">sage: (a, b, c, d, e) = F.gens()</span>
<span class="go">sage: d * b**2 * c**3</span>
<span class="go">b^2*c^3*d</span>
<span class="go">sage: F = AbelianGroup(3,[2]*3); F</span>
<span class="go">Multiplicative Abelian group isomorphic to C2 x C2 x C2</span>
<span class="go">sage: H = AbelianGroup([2,3], names="xy"); H</span>
<span class="go">Multiplicative Abelian group isomorphic to C2 x C3</span>
<span class="go">sage: AbelianGroup(5)</span>
<span class="go">Multiplicative Abelian group isomorphic to Z x Z x Z x Z x Z</span>
<span class="go">sage: AbelianGroup(5).order()</span>
<span class="go">+Infinity</span>
</pre></div>
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