Finite Group Theory

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 \(F\) 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 \(n\).

Let \(G\) be a finite group. Let \(V_1\) and \(V_2\) be two non-isomorphic irreducible complex representations. Let \(V = V_1 \oplus V_2\). Compute the algebra of linear transformations of \(V\) intertwining the action of \(G\): \[ \operatorname {Hom}_G(V,V). \]

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

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.

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

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 \(R\).

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?

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 \(A_5\)?

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

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 \[ \mathbb {F}_5(t)^G=\mathbb {F}_5(s). \]

Prove or disprove: each element of finite order in the general linear group \(\operatorname {GL}_n(\mathbb {C})\) is diagonalizable.

Suppose \((G,+)\) is a finite Abelian group of order \(3^3\cdot 2^3\). Suppose that \(6\cdot G=\{6x\mid x\in G\}\) has 6 elements. Classify \(G\) as a direct sum of cyclic groups.

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

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.

Determine, up to conjugacy, all \(3\times 3\) matrices \(M\) over \(\mathbb {Q}\) that satisfy \(M^3=2M^2\).

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 \(G\).

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.

Suppose \(G\) is a finite group with \(\operatorname {Aut}G\) solvable. Prove that \(G\) is solvable.

Suppose a finite group \(G\ne \{e\}\) has \(c\) conjugacy classes. Prove that \(G\) contains an element other than the identity of order at most \(c\).

Determine, up to conjugacy, all real \(3\times 3\) matrices \(A\) satisfying \(A^8=I\) and \(A^4\ne I\).

Prove that there is no simple group of order 132.

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 \(a,b,c,d\) integers such that \[ \det (A)\ne 0\qquad \text {where}\qquad A=\begin {pmatrix} a & b\\ c & d \end {pmatrix}. \] Let \(x\) and \(y\) be indeterminates, and set \(L=\mathbb {C}(x,y)\) and \(K=\mathbb {C}(x^a y^b,x^c y^d)\).

1. Prove that \(L\) is a finite extension of \(K\) and that \((L:K)=|\det (A)|\).
2. Show that \(L/K\) is Galois, and compute the Galois group.

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 \(n!\ge [G:N]\ge n\).

Prove that any group of order 345 is cyclic.

Prove that there is no simple group of order 192.

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 \(G\).

Determine, up to conjugacy, the elements of order 4 in \(GL_3(\mathbb {Q})\).

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, up to conjugacy, all \(4\times 4\) matrices \(M\) over \(\mathbb {Q}\) with \(M^5=-M^3\) and \(M^3\ne 0\).

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.

Determine representatives for the conjugacy classes in \(GL_3(\mathbb {F}_2)\).

Suppose \(G\) is a group acting transitively on a finite set \(S\), \(|S|\geq 2\). Prove that there exists an element \(\sigma \in G\) such that \(\sigma (s)\ne s\), for all \(s\in S\).

Let \(S^1\) be the circle group, i.e., the group of all complex numbers of norm 1 with multiplication. If \(A\) is a finite abelian group, define the dual group \(\widehat {A}\) to be the multiplicative group of all group homomorphisms \(A\to S^1\). Prove that \(A\cong \widehat {A}\).

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 \(G/H\).

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?

Recall that a Hermitian matrix is a complex matrix which equals its conjugate transpose. Determine the conjugacy classes of \(5\times 5\) Hermitian matrices \(A\) satisfying \(A^5+2A^3+3A=6I\).

Determine the number of conjugacy classes of \(4\times 4\) complex matrices satisfying \(A^3-2A^2+A=0\).

Let \(G\) be a finite \(p\)-group, with \(|G|>p\). Prove that the order of \(\operatorname {Aut}(G)\) is divisible by \(p\).

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.

Let \(E\subset F\) be a finite Galois extension of fields. Suppose that there is an element \(\alpha \) in \(F\) such that \(\alpha \notin E\) and such that \(\alpha \) is in every proper extension of \(E\) contained in \(F\). Show that the Galois group of \(F\) over \(E\) is cyclic of prime power order.

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

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

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?

Let \(S\) be a finite subset of \(\mathbb {C}\). Let \(X\) be the set of \(n\times n\) complex matrices all of whose eigenvalues lie in \(S\). Consider the conjugation action of \(\operatorname {GL}_n(\mathbb {C})\) acting on \(X\). Prove that this action has finitely many orbits. If \(n=3\), find a formula for the number of orbits in terms of the cardinality of \(S\).

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.

Let \(\mathbb {F}_q\) be a finite field of order \(q\). Compute the order of \(GL_n(\mathbb {F}_q)\), the group of invertible \(n\times n\) matrices with coefficients in \(\mathbb {F}_q\).

Let \(F\) be a field. Let \(M_5(F)\) denote the set of 5-by-5 matrices with entries in \(F\), and let \(\mathcal {N}\) denote the subset of nilpotent matrices, \[ \mathcal {N}:=\{A\in M_5(F)\mid \text {there exists }N\text { such that }A^N=0\}. \] Let \(X\) denote the subset of \(M_5(F)\) consisting of those matrices which are squares of nilpotent matrices, \[ X:=\{M\in M_5(F)\mid \text {there exists }A\in \mathcal {N}\text { such that }M=A^2\}. \] Observe that \(G=\operatorname {GL}(5,F)\) acts on \(X\) by conjugation. Determine, with proof, the number of orbits for the action of \(G\) on \(X\).

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

Let \(G\) be a finite group, let \(H\) be a subgroup of \(G\), and let \(\mathbb {C}[G]\) (respectively, \(\mathbb {C}[H]\)) denote the complex group algebra of \(G\) (respectively, \(H\)). Let \(I_H^G\) denote the functor from the category of finite-dimensional \(\mathbb {C}[H]\) modules to the category of finite-dimensional \(\mathbb {C}[G]\) modules which, on the level of objects, is defined by \[ I_H^G(M)=\mathbb {C}[G]\otimes _{\mathbb {C}[H]}M. \] Prove or disprove: \(I_H^G\) is an exact functor.

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 \(N\).

Let \(F\) be a field. Let \(X\) denote the set of 8-by-8 matrices \(A\) with entries in \(F\) such that \(A^3=0\) but \(A^2\ne 0\). Observe that \(G=\operatorname {GL}(8,F)\) acts on \(X\) by conjugation. Compute the number of orbits for the action of \(G\) on \(X\).

Let \(G\) be a finite group and let \(\mathbb {C}[G]\) denote the complex group algebra of \(G\). Show that the invariants functor which (on the level of objects) takes a \(\mathbb {C}[G]\) module \(M\) to \[ M^G:=\{m\in M\mid f\cdot m=m\text { for all }f\in \mathbb {C}[G]\} \] is an exact functor.

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 \(p^m\).

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

Let \(K\) be a finite field.

1. Show that \(|K|=p^n\) for some prime \(p\).
2. Show that \(K^*=K-\{0\}\) is a cyclic group with respect to multiplication.

Let \(F\) be a field such that the multiplicative group of \(F\) is cyclic. Prove that \(F\) is a finite field.

Set \[ J= \begin {pmatrix} 0&1&0&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0 \end {pmatrix} \in M_{10}(\mathbb {C}); \] here it is understood that unless otherwise noted, the entries of \(J\) are 0. Let \[ S=\{X\in M_{10}(\mathbb {C})\mid X^2=gJg^{-1}\text { for some }g\in \operatorname {GL}(10,\mathbb {C})\}. \] Observe that \(\operatorname {GL}(10,\mathbb {C})\) acts on \(S\) by conjugation. Compute the number of orbits for this action.

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.

Let \(G\) be a finite group with identity \(e\). Suppose that for all \(a,b\in G\) distinct from \(e\), there exists an automorphism \(\tau \) of \(G\) such that \(\tau (a)=b\). Prove that \(G\) is abelian.

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 \(G\).

Prove that a group of order 112 cannot be simple.

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 \(G\).

1. Suppose \(K\) is a field of prime order and \(E\) is the algebraic closure of \(K\). If \(F\) is any subfield of \(E\), prove that the multiplicative group of \(F\) is periodic, i.e., every element of the multiplicative group has finite order.
2. Suppose \(F\) is a field whose multiplicative group is periodic. Prove that for some prime \(p\), \(F\) is isomorphic to a subfield of the algebraic closure of the field of order \(p\).

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.

Let \(E\subset F\) be an extension of finite fields. Let \(q\) be the number of elements of \(E\), and let \(\phi \colon F\to F\) be defined by \(\phi (a)=a^q\). Prove that \(F\) is a Galois extension of \(E\) and that \(\phi \) generates its Galois group.

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 a finite abelian group is determined up to isomorphism by the number of elements of each order.

Let \(G\) be a finite group, and let \(C\) be the category of finitely generated \(\mathbb {Q}[G]\)-modules. Show the the functor which takes a \(\mathbb {Q}[G]\)-module to its submodule of invariants (that is, the set of \(m\in M\) such that \(gm=m\) for all \(g\in G\)) is exact.

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 finite group. Prove the following version of Schur’s Lemma: if \(\rho \colon G\to M_n(\mathbb {C})\) is a finite dimensional irreducible representation of \(G\) over the complex numbers, then every element of \(M_n(\mathbb {C})\) that commutes with \(\rho (g)\) for all \(g\in G\) is a scalar matrix.

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

Let \(G\) be a finite abelian group and let \(n\) be an integer. Show that the map which sends \(g\) to \(ng\) is a group homomorphism and that it is an isomorphism if and only if \(n\) is relatively prime to the order of \(G\).

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 \(x,y\in G\).]

Suppose \(F\) is a field and \(K=F(\alpha )\) is a finite Galois extension of \(F\). Assume that the Galois group of \(K\) over \(F\) is cyclic generated by an automorphism \(\sigma \) where \(\alpha ^\sigma =\alpha +1\). Prove that \(F\) has characteristic \(p\ne 0\) and that \(\alpha ^p-\alpha \in F\).

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 \(G\) be a finite group. Prove that every short exact sequence of \(\mathbb {C}[G]\)-modules splits, where \(\mathbb {C}[G]\) is the group algebra of \(G\) over the complex numbers.

Prove that a free group has no nonidentity elements of finite order.

If \(H\) is a finite group, prove that there exists a field \(k\) and a Galois extension of \(k\) with Galois group \(H\).

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 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 \(F\) be a field such that the multiplicative group of \(F\) is cyclic. Prove that \(F\) is a finite field.

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 \(p\).

Let \(K\subseteq L\) be finite fields. Prove that the extension \(L\) of \(K\) is Galois with cyclic Galois group.

Let \(f\colon \mathbb {Z}^n\to \mathbb {Z}^n\) be a map of free Abelian groups of rank \(n\) defined by a matrix \(A\) with non-zero determinant. Show that the cokernel of \(f\) is finite and that its order is equal to the absolute value of the determinant of \(A\).

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.

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

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

Prove that the group given by the presentation \[ \langle x,y\mid x^2=1,\ y^2=1,\ xyx=yxy\rangle \] has 6 elements.