Terminology: The group \(U_n\) is called the the group of (multiplicative) units in \(\Z_n\text{.}\) The function \(n\to |U_n|\text{,}\) important in number theory, is called the Euler phi function, written \(\phi(n)=|U_n|\text{.}\)
First, reduce \(x\) mod \(p\text{,}\) that is, write \(x=qp+r\) with \(0\leq r\leq p-1\text{.}\) Now consider two cases. The case \(r=0\) is trivial. If \(r\neq 0\text{,}\) apply the fact \(r^{|G|}=e\) (see Exercise 2.3.8) to the group \(G=U_p\text{.}\)
Show that \(s,t\) are either both even or both odd. The common evenness or oddness of \(s,t\) is called the parity of the permutation \(\sigma\text{.}\)
Notice that the index of the rightmost transposition in which the symbol \(a\) occurs has been reduced by 1 (from \(r\) to \(r-1\)). Finish this reasoning with an inductive argument.
Let \(\sigma\in S_n\) be written as a product of disjoint cycles. Show that the order \(\sigma\) is the least common multiple of the lengths of those disjoint cycles.
6.Alternative approach to multiplication in a factor group.
Given subsets \(S,T\) of a group \(G\text{,}\) define the set \(ST\) by
\begin{equation*}
ST = \{st\colon s\in S,t\in T\}.
\end{equation*}
Now suppose that \(H\) is a subgroup of \(G\text{.}\) Show that \((xH)(yH)=xyH\) for all \(x,y\) in \(G\) if and only if \(H\) is a normal subgroup of \(G\text{.}\)
Let \(K,H\) be groups, and let \(\phi\colon H\to \Aut(K)\) be a homomorphism. The semidirect product, denoted \(K\times_{\phi} H\text{,}\) or \(K\rtimes H\) if \(\phi\) is understood, is the set consisting of all pairs \((k,h)\) with \(k\in K\text{,}\)\(h\in H\) 1
The notation \(K\times H\) is understood to be the direct product group, so we do not use the Cartesian product notation to describe semidirect product, to avoid confusion, even though the underlying set for the direct product and the semidirect product are in fact the same Cartesian product \(K\times H\text{.}\)
with the group multiplication operation \(\ast\) given by
Two examples demonstrate why this is a useful construction. The dihedral group \(D_n\) is (isomorphic to) the semidirect product \(C_n\rtimes C_2\text{,}\) where \(C_n\) is the cyclic group generated by the rotation \(R_{1/n}\) (rotation by \(1/n\) of a revolution) and \(C_2\) is the two-element group generated by any reflection \(R_L\) in \(D_n\text{.}\) The map \(\phi\colon C_2 \to \Aut(C_n)\) is given by \(F_L \to
[R_{\theta} \to R_{-\theta}]\text{.}\) The Euclidean group of congruence transformations of the plane is (isomorphic to) the group \(\R^2\rtimes O(2)\text{,}\) where \((\R^2,+)\) is the additive group of \(2\times 1\) column vectors with real entries, and \(O(2)\) is the group of \(2\times 2\) real orthogonal matrices. The map \(\phi\colon O(2)\to \Aut(\R^2)\) is given by \(g\to [v\to gv]\text{,}\) that is to say, the natural action of \(O(2)\) on \(\R^2\text{.}\) [The Euclidean group element \((v,g)\) acts on the point \(x\in\R^2\) by \(x\to
gx+v\text{.}\)]
Do all the necessary details to show that \(K\rtimes H\) is indeed a group.
(Characterization of semidirect products) Suppose that \(K,H\) are subgroups of a group \(G\text{.}\) Let \(KH=\{kh\colon k\in K,h\in
H\}\text{.}\) Suppose that \(K\) is a normal subgroup of \(G\text{,}\) that \(G=KH\text{,}\) and that \(K\cap H=\{e\}\text{.}\) Show that \(\phi\colon H\to \Aut(K)\text{,}\) given by \(\phi(h)(k)=hkh^{-1}\text{,}\) is a homomorphism. Show that \(\psi\colon K\times_\phi H\to G\text{,}\) given by \(\psi(k,h)=kh\text{,}\) is an isomorphism.
Suppose that \(\phi\colon H\to \Aut(K)\) is the trivial homomorphism (that is, \(\phi(h)\) is the identity homomorphism on \(K\text{,}\) for all \(h\in H\)). Show that \(K\times_{\phi} H\approx K\times H\) in this case.
