Symmetric and Alternating Groups

Let \(F\) be a field of characteristic \(p\). Let \(S_n\) be the group of permutations of \(X = \{1,2,\ldots ,n\}\). Let \(V = F[X]\) be the space of \(F\)-valued functions. It is a linear representation of \(S_n\), where \(\sigma \in S_n\) acts on \(f \in F[X]\cong F^n\) by \((\pi (\sigma )f)(x)=f(\sigma ^{-1}x)\). We have an intertwining map \(P:F[X]\to F\), where on \(F\) we have the trivial action of \(S_n\), given by \(f\mapsto f(1)+f(2)+\cdots +f(n)\). Prove that this map splits, that is, there exists an intertwining map \(Q\) in the opposite direction such that \(P\circ Q=1\), if and only if \(p\) does not divide \(n\).

Prove that the alternating group \(A_5\) is the only simple group of order 60.

Find a complete list of irreducible representations of the alternating group \(A_4\).

Prove that the alternating group \(A_5\) is simple.

Prove that the Galois group of an irreducible polynomial \(f(x) \in \mathbb {Q}[x]\) of degree \(p\) (a prime) with one pair of complex conjugate roots and \(p-2\) real roots is the full symmetric group \(S_p\).

Prove that each 7-Sylow subgroup of the symmetric group \(S_{15}\) is abelian.

Recall that the standard representation of \(S_n\) is the subspace

with \(S_n\) acting by permutation of the coordinates. Show that \(\wedge ^2 V_5\) is an irreducible representation of \(S_5\).

Suppose \(\sigma \) and \(\tau \) are two elements of \(A_5\) such that, for every finite dimensional complex representation \(\rho \) of \(S_5\), \(\chi _\rho (\sigma )=\chi _\rho (\tau )\) (here \(\chi _\rho \) denotes the character of \(\rho \)). Are \(\sigma \) and \(\tau \) conjugate in \(A_5\)?

Is every automorphism of the alternating group \(A_4\) an inner automorphism?

Let \(S_n\) denote the symmetric group on \(n\) elements. Does there exist an injective homomorphism \(S_6\longrightarrow S_5\times S_5\)?

Prove that there is no element in \(S_9\) of order 18.

How many elements of \(S_7\) commute with the permutation \((123)(45)(67)\)?

Determine the integers \(n\) for which there exists a surjective homomorphism of symmetric groups \(S_{n+1}\longrightarrow S_n\).

Let \(k\) be a field and \(k(x)\) the field of rational functions over \(k\), in the indeterminate \(x\). Consider the automorphisms \(\sigma \) and \(\tau \) of \(k(x)/k\) defined by \[ \sigma (x)=\frac {1}{1-x}\qquad \text {and}\qquad \tau (x)=\frac {1}{x}. \]

1. Prove that the subgroup \(G=\langle \sigma ,\tau \rangle \) of \(\operatorname {Aut}(k(x)/k)\) is isomorphic to \(S_3\).
2. Prove that the fixed subfield under the action of \(G\) equals \(k(t)\), where \[ t=\frac {(x^2-x+1)^3}{x^2(x-1)^2}. \]

Let \(n\) a positive integer, and let \(\mathbf {x}:=x_1,\ldots ,x_n\) be indeterminates over \(\mathbb {Q}\). For each positive integer \(i\) set \[ \alpha _i=x_1^i+\cdots +x_n^i. \]

1. Clearly \(\mathbb {Q}(\alpha _1,\ldots ,\alpha _n)\subseteq \mathbb {Q}(s_1,\ldots ,s_n)\), the field of symmetric functions in the \(\mathbf {x}\). What is the degree of this extension?
2. Prove the characteristic polynomial of an \(n\times n\) matrix \(A\) over \(\mathbb {Q}\) is completely determined by the elements \(\operatorname {trace}(A),\ldots ,\operatorname {trace}(A^n)\).

How many elements of the symmetric group \(S_n\) commute with a fixed \(k\)-cycle in \(S_n\)?

Let \(\sigma \) be a 5-cycle in \(S_5\). How many elements of \(S_5\) commute with \(\sigma \)? How many elements of the subgroup \(A_5\) commute with \(\sigma \)? How many conjugacy classes of 5 cycles are there in \(A_5\)?

Given an integer \(n\), prove that there exists a monic polynomial with integer coefficients whose Galois group over \(\mathbb {Q}\) is \(S_n\).

Let \(f(x)\in \mathbb {Q}[x]\) be a polynomial of degree \(n\ge 4\) and let \(K\) be a splitting field of \(f\) over \(\mathbb {Q}\). Suppose that \(\operatorname {Gal}(K/\mathbb {Q})\) is the symmetric group \(S_n\). If \(\alpha \in K\) is a root of \(f(x)\), show that \(\alpha ^n\notin \mathbb {Q}\).

Let \(A\) be a real \(n\times n\) matrix. We say that \(A\) is a difference of two squares if there exist real \(n\times n\) matrices \(B\) and \(C\) with \(BC=CB=0\) and \(A=B^2-C^2\).

1. If \(A\) is a diagonal matrix, show that it is a difference of two squares.
2. If \(A\) is a symmetric matrix that is not necessarily diagonal, again show that it is a difference of two squares.
3. Suppose \(A\) is a difference of two squares, with corresponding matrices \(B\) and \(C\) as above. If \(B\) has a nonzero real eigenvalue, prove that \(A\) has a positive real eigenvalue.

Let \(p\) be a prime. Show that an element in the symmetric group \(S_n\) has order \(p\) if and only if it is a product of commuting \(p\)-cycles. Show by an explicit example that this need not be the case if \(p\) is not prime.

Show that for any positive integer \(n\), every element of order 2 in the alternating group \(A_n\) is the square of an element of order 4 in the symmetric group \(S_n\).

Compute the number of elements of order 4 in the symmetric group \(S_7\).

Consider the symmetric group \(S_p\) for \(p\) an odd prime. Let \(C\) be the centralizer of the \(p\)-cycle \(\sigma =(1\ 2\ \ldots \ p)\) and let \(N\) be the normalizer of \(\langle \sigma \rangle \) in \(S_p\). What is the order of each of the groups \(C\) and \(N\)? Are they abelian?

Let \(\omega =e^{2\pi i/3}\). Let \(\sigma \) and \(\tau \) be automorphisms of \(\mathbb {C}(x)\) which fix \(\mathbb {C}\) and satisfy \[ \sigma (x)=\omega x,\qquad \tau (x)=1/x. \] Prove that the group \(\langle \sigma ,\tau \rangle \) is isomorphic to the symmetric group \(S_3\), and determine the subfield of \(\mathbb {C}(x)\) that is fixed by this group.

Determine, with proof, the maximal order of an element in a Sylow 3-subgroup of the symmetric group on 9 letters.

Describe the structure of a Sylow 2-subgroup of the symmetric group \(S_n\) in terms of products and semidirect products of known groups for:

1. \(n=4\); and
2. \(n=8\).

Determine the number of subgroups of order 16 in the symmetric group \(S_6\).

Prove that the Galois group of \(x^3+x-1\) over \(\mathbb {Q}\) is the symmetric group on three letters.

If \(G\) is a simple group of order 60, prove that \(G\) is isomorphic to \(A_5\), the alternating group of degree 5.

Describe the 2-Sylow subgroup of \(S_6\) (that is, show that it is isomorphic to a group described in terms of known groups).

Prove that the symmetric group \(S_n\) on \(n\) elements is generated by the transpositions \((12),(23),\ldots ,(n-1\ n)\).

Prove that the Galois group of \(X^3-X-1\) over \(\mathbb {Q}\) is \(S_3\), the symmetric group on three elements.

Let \(A_n\) be the alternating group of degree \(n\) (i.e., \(n\) is the number of objects permuted by \(A_n\)). If \(n\ge 5\) and \(H\) is a subgroup of \(A_n\) such that the order of \(H\) is \(\frac {(n-1)!}{2}\), then prove that \(H\) is isomorphic to \(A_{n-1}\).

Suppose \(G\) is a finite simple group such that \(G\) has exactly 5 distinct Sylow \(p\)-subgroups for some prime \(p\). Prove that \(G\) is isomorphic to \(A_5\), the alternating group of order 60.