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Magazine Problem 6
Show that the mapping \(\phi: S_{3} \rightarrow S_{3}\) defined by letting \(\phi(x)=x^{-1}\) for all \(x \in S_{3}\) is not an automorphism of \(S_{3}\).
Problem 7
Let \(G\) be a group, \(H \triangleleft G, \phi \in \operatorname{Aut}(G)\). Show that \(\phi(H) \triangleleft G\).
Problem 7
Find the order of the indicated element in the indicated quotient group. $$ 3+\langle 8\rangle \text { in } \mathbb{Z}_{12} /\langle 8\rangle $$
Problem 7
Determine whether the indicated subgroup is normal in the indicated group. $$ K=\left\\{\rho_{0},(12)(34),(13)(24),(14)(23)\right\\} \text { in } S_{4} $$
Problem 8
Let \(H=\left\\{\phi \in S_{n} \mid \phi(n)=n\right\\}\). Find the index of \(H\) in \(S_{n}\).
Problem 8
Determine whether the indicated subgroup is normal in the indicated group. $$ \langle(123)\rangle \text { in } S_{4} $$
Problem 8
Find the order of the indicated element in the indicated quotient group. $$ 3+\langle 6\rangle \text { in } \mathbb{Z}_{15} /\langle 6\rangle $$
Problem 9
Show that \(\operatorname{Inn}\left(S_{3}\right) \cong S_{3}\).
Problem 9
Find the order of the indicated element in the indicated quotient group. $$ 2+\langle 6\rangle \text { in } \mathbb{Z}_{15} /\langle 6\rangle $$
Problem 9
Let \(H\) be a subgroup of a group \(G\). For any \(a, b \in G,\) let \(a \sim b\) if and only if \(a b^{-1} \in H\). Show that the relation \(\sim\) so defined is an equivalence relation on \(G,\) with equivalence classes the right cosets \(\mathrm{Ha}\) of \(\mathrm{H}\).
