Sylow Theorems

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

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

Show there is no simple group of order \(2304 = 2^8 \cdot 3^2 = 256 \cdot 9\).

Find a complex number \(\eta \) such that \(\mathbb {Q}(\eta )/\mathbb {Q}\) is Galois with Galois group isomorphic to the dihedral group of order 8.

Is every group of order 480 solvable?

Show there is no simple group of order \(448=2^6\cdot 7\).

Show that there is no simple group of order \(552=2^3\cdot 3\cdot 23\).

Suppose that \(p\) is the smallest prime dividing the order of a group \(G\), and that \(P\) is a Sylow \(p\)-subgroup of \(G\). If \(P\) is cyclic, show that \[ N_G(P)=C_G(P), \] i.e., that the normalizer of \(P\) in \(G\) agrees with the centralizer of \(P\) in \(G\).

Show there is no simple group of order 144.

How many elements of order 3 can a group of order 21 have? List all the possibilities.

Show that there is no simple group of order 112. You may use the fact that \(A_7\) is simple.

Prove that there is no simple group of order 132.

Prove that any group of order 345 is cyclic.

Prove that there is no simple group of order 192.

Prove that every group of order 15 is cyclic.

What is the number of elements of order 11 in a simple group of order 660?

Let \(P\) be a \(p\)-Sylow subgroup of a group \(G\), and \(N\) a normal subgroup of \(G\). Prove that \(P\cap N\) is a \(p\)-Sylow subgroup of \(N\).

Determine the number of Sylow \(p\)-subgroups of \(GL_2(\mathbb {F}_p)\).

Prove that the number of Sylow \(p\)-subgroups of \(GL_2(\mathbb {F}_p)\) is \(p+1\).

Determine the number of 5-Sylow subgroups of \(\operatorname {SL}_2(\mathbb {F}_5)\).

Let \(G\) be a group of order \(p^2q\), where \(p\) and \(q\) are distinct primes. Show that \(G\) has a normal Sylow subgroup.

Let \(G\) be a group of order \(p^n\), where \(p\) is a prime number and \(n>0\) is an integer. Prove that the center of \(G\) is not trivial.

A group of order 81 acts on a set \(X\) with 30 elements. Show that some element of \(X\) is fixed by at least 27 elements. How many elements of \(X\) are fixed by precisely 3 elements, assuming there is at least one such element?

Show that a finite group of order 105 has a non-trivial normal subgroup.

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. \(n=8\).

Let \(G\) be a group of order 63. Show that \(G\) contains an element of order 21.

Suppose \(G\) is a finite group of order a power of a prime. If \(H\) is a nonidentity normal subgroup of \(G\), prove that \(H\) contains a nonidentity element of \(Z(G)\). \[ Z(G)=\{x\in G\mid xy=yx\text { for all }y\in G\}. \]

Prove that a group of order 112 cannot be simple.

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

Suppose \(G\) is a group of order \(p^n\) where \(p\) is a prime and \(n>1\). If \(A\) is the group of all automorphisms of \(G\), prove that \(p\) divides the order of \(A\).

Let \(G\) be a group of order \(p^n\), where \(p\) is a prime number and \(n>0\) is an integer. Show that the center of \(G\) is not trivial.

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

Let \(G\) be a nonabelian group of order \(p^3\) for a prime \(p\) and and let \(Z\) be its center. Show that \(G/Z\cong \mathbb {Z}/p\mathbb {Z}\oplus \mathbb {Z}/p\mathbb {Z}\).

If \(G\) is a group of order 105, prove that \(G\) has a normal cyclic subgroup of order 35.

Let \(G\) be a nonabelian group of order \(p^3\) where \(p\) is a prime. Prove that \(Z(G)\) and \(G'\) are equal and have order \(p\). [Here \(Z(G)\) is the center of \(G\) and \(G'\) is the commutator subgroup of \(G\), i.e., \(G'\) is generated by all elements of the form \(x^{-1}y^{-1}xy\) with \(x,y\in G\).]

Let \(G\) be a group of order 55 acting on a set \(X\) consisting of 24 elements. Show that there is at least one element fixed by every element of \(G\).

Let \(p\) be a prime number, and let \(G\) be a finite group of order \(p^n\) acting on a set \(X\) consisting of \(r\) elements. Show that the number of points fixed by all elements of \(G\) is congruent to \(r\) modulo \(p\).

Let \(G\) be a finite group of order \(p^n\) acting on a set \(X\) consisting of \(k\) elements. Show that the number of points fixed by all elements of \(G\) is congruent to \(k\) modulo \(p\).

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.

1. No group of order 156 is simple.
2. Every group of order 289 is abelian.