Normal Subgroups and Commutator Subgroups

Let \(\mathbb {Q}\) be the set of rational numbers, considered a group with respect to the usual addition. Prove that \(\mathbb {Q}\) is indecomposable, that is, it is not isomorphic to a direct sum \(A \oplus B\) of two non-trivial subgroups.

Let \(n\) and \(m\) be two relatively prime natural numbers. Let \(N = nm\). Let \((G,+)\) be an abelian group such that \(0 = Ng = g+\cdots +g\) (\(N\) summands) for every \(g \in G\). Let \(G_n=\{g\in G \mid ng=0\}\) and \(G_m=\{g\in G \mid mg=0\}\). Then \(G_n\) and \(G_m\) are subgroups of \(G\). (You do not need to verify that.) Prove that \(G\) is isomorphic to \(G_n \oplus G_m\).

Prove that if \(p\) is prime, then each group with \(p^d\) elements has a non-trivial center.

Let \(G\) be a group, and \(N\) the subgroup generated by all elements of the form \(g^2\) for \(g \in G\). Prove that \(N\) is a normal subgroup containing the commutator subgroup of \(G\).

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

Find a non-abelian finite subgroup of \(\operatorname {GL}_2(\mathbb {C})\) that is not conjugate to a finite subgroup of \(\operatorname {GL}_2(\mathbb {Z})\). Here for a ring \(R\), \(\operatorname {GL}_2(R)\) denotes the group of invertible \(2 \times 2\) matrices with entries in \(R\).

Suppose \(G\) is a finitely generated Abelian group such that the automorphism group of \(G\) is finite. Let \(T\) be the torsion subgroup of \(G\). Describe all possibilities for \(G/T\) up to isomorphism.

Let \(\sigma \) be the automorphism of \(\mathbb {F}_5(t)\) sending \(t\) to \(t+1\). Let \(G\) be the subgroup of \(\operatorname {Aut}(\mathbb {F}_5(t))\) generated by \(\sigma \). Find an element \(s\in \mathbb {F}_5(t)\) such that \[ \mathbb {F}_5(t)^G=\mathbb {F}_5(s). \]

Suppose \(\varphi :G\longrightarrow H\) is a group homomorphism and \(H\) is a group of order 33. Further suppose that \(N\supseteq \ker \varphi \) is a subgroup of \(G\). Prove that \(N\) is normal.

Let \(G\) be a finite simple group with identity \(e\). Suppose that \(A\) and \(B\) are distinct maximal proper subgroups of \(G\). If \(A\) and \(B\) are abelian, prove that \(A\cap B=\{e\}\).

Hint: Prove that \(A\cap B\) is normal in \(G\).

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 \(G\).

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 \(G\) be a finite \(p\)-group, for \(p\) a prime integer. Let \(e\) denote the identity element, and \(Z\) the center of \(G\). If \(N\ne \{e\}\) is normal, prove that \(N\cap Z\ne \{e\}\).

Let \(G\) be a nonabelian simple group. Prove that \(\operatorname {Aut}G\) contains a normal subgroup isomorphic to \(G\).

Let \(G\) be a group (not necessarily finite), and suppose that \(H\) is a subgroup of index \(n\). Show that there is a normal subgroup \(N\) of \(G\) with \(n!\ge [G:N]\ge n\).

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 \(A_5\)?

Let \(H\) be a proper subgroup of a finite group \(G\). Prove that \(\displaystyle \bigcup _{g\in G}gHg^{-1}\) does not equal \(G\).

Set \(G=GL_2(\mathbb {C})\). Construct a proper subgroup \(H\) of \(G\) such that \(\displaystyle \bigcup _{g\in G}gHg^{-1}\) equals \(G\).

Let \(G\) be a group of order \(p^3\), where \(p\) is a prime. Suppose \(G\) is not abelian. Show that the center \(Z\) of \(G\) is isomorphic to \(\mathbb {Z}/(p)\) and that \(G/Z\cong \mathbb {Z}/(p)\times \mathbb {Z}/(p)\).

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 \(GL_2(\mathbb {F}_p)\).

Show that \((\mathbb {Q}/\mathbb {Z},+)\) has one and only one subgroup of order \(n\), for each integer \(n\ge 1\), and that this subgroup is cyclic.

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

Let \(G\) be a \(p\)-group with \(|G|>p\). Show that

1. \(G\) has nontrivial center;
2. \(G\) has a normal subgroup of every order \(p^m<|G|\).

Let \(G\) be a finite abelian group and \(H\) a subgroup of \(G\). Show that \(G\) has a subgroup isomorphic with \(G/H\).

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

Let \(G\) be the subgroup of \(\operatorname {GL}_2(\mathbb {R})\) consisting of matrices of the form \[ \begin {pmatrix} a & c\\ 0 & b \end {pmatrix}. \] Is \(G\) solvable?

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.

Show that every finitely generated subgroup of \(\mathbb {Q}/\mathbb {Z}\) is cyclic.

Let \(U\) be the subgroup of \(\operatorname {GL}_2(\mathbb {Z}/p\mathbb {Z})\) consisting of upper triangular matrices. Is \(U\) solvable? Prove or disprove.

Let \(L=\mathbb {Q}(e^{2\pi i/8})\). Determine all subgroups of the Galois group \(\operatorname {Gal}(L/\mathbb {Q})\), and the corresponding intermediate fields.

Let \(p\) be a prime integer. Let \(G\) be a finite \(p\)-group, with identity \(e\), and center \(Z\). If \(N\ne \{e\}\) is a normal subgroup of \(G\), prove that \(N\cap Z\ne \{e\}\).

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.

Let \(N\) be a normal subgroup of a finite group \(G\). Assume that the center of \(N\) is trivial and that any automorphism of \(N\) is inner. Show that there exists a normal subgroup \(H\) of \(G\) such that \(G\) is isomorphic to the direct product of \(H\) and \(N\).

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 (up to isomorphism) all groups \(G\) which satisfy the following condition: there exists a proper subgroup \(H\) of \(G\) which contains all proper subgroups of \(G\).

Let \(G\) be a finite non-abelian group of order \(p^3\) where \(p\) is a prime number. If \(Z\) is the center of \(G\), show that \(G/Z\) is isomorphic to \(\mathbb {Z}/p\mathbb {Z}\oplus \mathbb {Z}/p\mathbb {Z}\).

Show that \[ \mathbb {Q}/\mathbb {Z}\cong \bigoplus _p(\mathbb {Q}/\mathbb {Z})_{(p)} \] where \(p\) runs over all primes numbers and \((\mathbb {Q}/\mathbb {Z})_{(p)}\) denotes the subgroup of elements of \(\mathbb {Q}/\mathbb {Z}\) whose order is a power of \(p\).

Let \(G\) be a finite group and \(p\) a prime number such that \(|G|=p^n\) for some \(n>0\). Prove that \(G\) has nontrivial center and that for all integers \(0<m<n\), \(G\) has at least one normal subgroup of order \(p^m\).

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

Let \(R\) be a commutative ring with identity. Let \(G\) be a finite subgroup of \(R^\times \), the group of units of \(R\). Prove that if \(R\) is an integral domain, then \(G\) is cyclic.

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 \(Z(G)\). \[ Z(G)=\{x\in G\mid xy=yx\text { for all }y\in G\}. \]

Suppose \(H\) is a subgroup of the finite group \(G\) and the index of \(H\) in \(G\) is the smallest prime dividing the order of \(G\). Prove that \(H\) is a normal subgroup of \(G\).

Suppose \(H\) is a subgroup of the group \(G\) such that \(x^2\in H\) for all \(x\in G\). Prove that \(H\) is a normal subgroup of \(G\).

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 finite group, and let \(H\) be a subgroup of index \(p\), where \(p\) is the smallest prime number dividing the order of \(G\). Show that \(H\) is a normal subgroup of \(G\).

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 \(H\) be a proper subgroup of the finite group \(G\). Prove that there is an element \(x\) in \(G\) such that \(x\) is not conjugate in \(G\) to any element of \(H\).

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 finite group with the property that for each positive integer \(n\), there are at most 2 subgroups of order \(n\) in \(G\). Prove that \(G\) is cyclic.

Let \(H\) be a proper subgroup of the finite group \(G\). Prove that \(G\) has a conjugacy class \(C\) such that \(C\) contains no element of \(H\).

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 \(A_{n-1}\).

Let \(G\) be a finite subgroup of the group \(\operatorname {GL}_n(\mathbb {C})\) of invertible \(n\) by \(n\) matrices with entries in \(\mathbb {C}\).

1. Show that every element of \(G\) is diagonalizable over \(\mathbb {C}\).
2. Give an example of a finite subgroup of \(\operatorname {GL}_n(\mathbb {Q})\) which contains elements which are not diagonalizable over \(\mathbb {Q}\).

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. Let \(H\) be a maximal subgroup of the finite group \(G\) and let \(M\) be a minimal normal subgroup of \(G\). Assume that \(M\) is abelian but \(H\) does not contain \(M\). Prove that \(G=HM\), \(H\cap M=\{1\}\) (the identity subgroup), and \(|G:H|\) (the index of \(H\) in \(G\)) is a power of a prime.
2. Use the result in (a) to show that that if \(G\) is a finite solvable group and \(H\) is a maximal subgroup of \(G\), then \(|G:H|\) is a power of a prime.

Assume that \(A\) and \(B\) are solvable subgroups of the group \(G\) and \(G=AB\). Here \(AB=\{ab\mid a\in A,\ b\in B\}\).

1. Give an example showing that \(G\) does not have to be solvable.
2. Prove that if either \(A\) or \(B\) is a normal subgroup of \(G\), then \(G\) is solvable.

A finite subgroup of the multiplicative group of nonzero elements of a field is cyclic.