91Ó°ÊÓ

Prove that every finite \(p\)-group is solvable.

Show that a Sylow 2-subgroup of \(S_{6}\) is isomorphic to \(D_{8} \times I_{2}\)

To isomorphism, how many abelian groups are there of order \(288 ?\)

(i) Prove that the composite of two reflections in Isom \(\left(\mathbb{R}^{2}\right)\) is either a rotation or a translation. (ii) Prove that every rotation is a composite of two reflections. Prove that every translation is a composite of two reflections. (iii) Prove that every isometry \(\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) is a composite of at most three reflections.

(i) Let \(G\) be a finite group and let \(P\) be a Sylow \(p\)-subgroup of \(G\). If \(H \triangleleft G\), prove that \(H P / H\) is a Sylow \(p\)-subgroup of \(G / H\) and \(H \cap P\) is a Sylow \(p\)-subgroup of \(H\). (ii) Let \(P\) be a Sylow \(p\)-subgroup of a finite group \(G\). Give an example of a subgroup \(H\) of \(G\) with \(H \cap P\) not a Sylow \(p\)-subgroup of \(H_{\text {. }}\)

(i) If \(\rho\) is a reflection in \(O_{2}(\mathbb{R})\), prove that there is a rotation \(R \in O_{2}(\mathbb{R})\) with \(R \rho R^{-1}=\sigma\), where \(\sigma(z)=\bar{z}\). (ii) If \(G\) is a subgroup of \(O_{2}(\mathbb{R})\) containing a reflection \(\rho\), prove that there is a rotation \(R \in\) Isom \(\left(\mathbb{R}^{2}\right)\) with \(R G R^{-1}\) containing complex conjugation.

Prove that there are no simple groups of order \(300,312,616\), or 1000 .

Prove that the additive group \(Q\) is not a direct sum: \(Q \neq A \oplus B\), where \(A\) and \(B\) are nonzero subgroups.

Prove that if every Sylow subgroup of a finite group \(G\) is normal, then \(G\) is the direct product of its Sylow subgroups.

Let \(G\) be a finite abelian group. Prove that if \(x \in G\) has maximal order (that is, if \(x\) has order \(n\), then there is no element in \(G\) having larger order), then \(\langle x\rangle\) is a direct summand of \(G\).