Permutation group

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Problem: Solve the equation in the permutation group \( S_{\mathcal{T}}:(1234567) \cdot X \cdot(26)=(134) \cdot(275) \). Expand the permutation into cycles and transpositions.

1.3.1 Permutation group

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Problem: Find the sign and the order of permutation \( \alpha \), and also find the permutation \( \alpha^{1644} \), where \[ \alpha=(02) \cdot(0982) \cdot(45) \cdot(3579) \cdot(2684) \cdot(1825) \text {. } \]

1.3.2 Permutation group

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Problem: Find the number of distinct subgroups of order 7 in the permutation group \( S_{15} \).

1.3.3 Permutation group

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Problem: Is the subgroup \( \left\{\sigma \in S_{n} \mid \sigma(1)=1\right\} \) a normal divisor in \( S_{n} \) ?

1.3.6 Permutation group

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Problem: In the symmetrical group \( S_{5} \) find out if the set \( \{(12),(123),(1234)\} \) is a coset with respect to some subgroup.

1.3.4 Permutation group

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Problem: Let \( A \) - be some vertex of the regular tetrahedron. Prove that the set of all selfcoincidences of the tetrahedron that leave the point \( A \) fixed, is a group isomorphic to the group \( S_{3} \).

1.3.5 Permutation group

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Problem: Given is the goup \( G=(a, b) \) and its subgroup \( H \). 1. Find the order of elements \( a, b, a b \). 2. Determine the order \( H \) and make a Cayley table for it, 3. Check that \( b H=H b \), and deduce from this that the subgroup \( H \) is normal in \( G \). 4. Describe the cosets \( G \) by \( H \). 5. Check that \( G / H \) is cyclic, and find its order. 6. Determine the order of the group \( G \). 7. Determine whether a cyclic subgroup with generator \( b \) is normal in \( G \), 8. Find all subgroups \( Z(b) \). \[ a=\left(\begin{array}{lll} 1 & 2 & 3 \end{array}\right), \quad b=\left(\begin{array}{ll} 12 \end{array}\right)(45), \quad H=(a) \text {. } \]

1.3.7 Permutation group

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Problem: Do permutations of order 11 generate a group \( S_{11} \) ?

1.3.8 Permutation group

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Problem: Prove that the group \( A_{5} \) is simple, that is it has no proper normal subgroups.

1.3.9 Permutation group

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Problem: Let \( G- \) a group, \( |G|=6 \). Prove that \( G \) is commutative, or \( G \cong S_{3} \) (is isomorphic).

1.3.10 Permutation group

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Problem: Prove that all elements of order 11 are conjugate in \( S_{11} \). Note: the element \( b \) of the group \( G \) is conjugate of \( a \) through \( g \), if \( b=g a g^{-1} \).

1.3.11 Permutation group

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Problem: Find all elements of the group \( S_{n} \), permutable with the cycle \( \left(\alpha_{1} \alpha_{2} \ldots \alpha_{n}\right) \), where \( \left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}\right) \) - the permutation of numbers \( 1,2, \ldots, n \).

1.3.12 Permutation group

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Problem: Prove that if the group \( G \) is non-commutative and \( |G|=8 \), then \( G \cong D_{4} \) or \( G \cong Q_{8} \).

1.3.13 Permutation group

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Problem: Prove that Aut \( S_{3}=\operatorname{Inn} S_{3} \cong S_{3} \).

1.3.14 Permutation group

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Problem: Prove that two permutations are conjugate in a group \( S_{n} \) if and only if they have the same cycle structure (i.e. their decomposition into products of independent cycles for any \( k \) contains the same number of cycles of length \( k \) ).

1.3.15 Permutation group

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