FURTHER EXERCISES FOR CHAP. II
(For 10 and 11 consult Exx. 15 and 16 at the end of Chap. I.)
10. Let 
be an abstract group with elements a1, …, x1, …. Let Σ be a system of groups such that for each member ′ of Σ there is a given an isomorphism i(′) between and ′. Show that the set of all one-to-one correspondences c(′, ″, a) between any ′ and any ″ which map the element i(′)x of ′ onto the element i(″)ax of ″ is a groupoid 
(Σ, ).
11. Let be a groupoid. For any two units e, e′ linked by x show that ex = x = xe′. Furthermore, show that the mapping of a onto x−1ax is an isomorphism between e and e′. Let Σ be the system of the groups e, e′, … attached to the units e, e′, … of . Let be an abstract group mapped by isomorphisms i(e), i(e′), … onto e, e′, … Show that is isomorphic to (Σ, 
).
12. Every homomorphism h of a multiplicative domain 
into another multiplicative domain defines on the normal multiplicative congruence relation: a R(h)b if and only if ha = hb. Conversely, if R is a normal multiplicative congruence relation on , then the residue classes form a multiplicative domain /R according to the rule of multiplication 
where 
denotes the residue class modulo R represented by the element x of . The mapping h that maps x onto is a homomorphism (natural homomorphism) of onto /R which induces on the congruence relation R in the sense defined above. If j is another homomorphism of inducing the normal multiplicative congruence relation R = R (j) on , then j induces the isomorphism between /R(j) and j () which maps onto jx.
13. Show that the intersection of two systems of imprimitivity of a transitive permutation group is itself a system of imprimitivity provided the intersection contains more than one letter.
14. (G. E. Wall.) In the ring I8 consisting of the eight residue classes of the rational integers modulo 8 show that the mappings (z, az + b) that map the element z of I8 onto the element az + b (a odd; a, b contained in I8) form a group of order 32. Show that the mapping of (z, az + b) onto (z, az + (b + (a2— 1)/2)) is an outer automorphism of that maps each element of onto a conjugate element under .
15. If a normal subgroup of a group and its factor group both are solvable, then is solvable.
16. The product of a solvable normal subgroup of a group and a solvable subgroup of is a solvable subgroup of . (Use Ex. 15.)
17. The radical R() of a group is defined as a solvable normal subgroup of which is not contained in a larger solvable normal subgroup of (maximal solvable normal subgroup). Show that there is at most one radical of and that it is a characteristic subgroup. If the maximal condition is satisfied for the solvable normal subgroups of (, then has a radical. (Use Ex. 16.)
13 7297 Zassenhaus, Theory of Groups
18. If the group has a radical R(), then the radical of each normal subgroup 
of is equal to the intersection of and the radical of . If is solvable, then R(/) = R()/; in particular R(/R()) = R()/R().
19. Each k-step metabelian subgroup of a group is contained in a maximal k-step metabelian subgroup of , i. e., a k-step metabelian subgroup not contained in a larger k-step metabelian subgroup.
20. If there are maximal solvable subgroups of a group , then the radical of is the intersection of the maximal solvable subgroups.
21. If, for a fixed k, every solvable subgroup of a group is at most k-step metabelian, then every solvable subgroup is contained in a maximal solvable subgroup, and has a radical.
22. Show that for any homomorphism h of a group onto a group 
we have h((a, b)) = (ha, hb) for a, b contained in and that h(Dr()) = Dr(h()) for r = 0, 1, 2, …
23. Let 
be a semi-group with unit element.
a) The element a is called a left divisor (right divisor) of the element b of if there is an equation b = ax (b = ya), where x and y respectively occur in . Show that this relation is reflexive and transitive.
b) Two elements are called left equivalent (right equivalent) if each is a left divisor (right divisor) of the other. Show the normality of this relation. Show that equivalent elements can be substituted in one-sided divisor relations.
c) If a is a left divisor of b, then ca is a left divisor of cb. If a is a right divisor of b, then ad is a right divisor of bd.
d) We say a divides b if there are equations
where all factors belong to . Show that the relation a divides b is reflexive and transitive. If a is a left divisor or a right divisor of b, then a divides b. If a divides b and a' divides then aa' divides bb'.
e) We say a is equivalent to b if a divides b and b divides a. Show thatordering relations defining a poset this equivalence relation is normal. Show that equivalent elements can be substituted in divisor relations.
f) Interpret each of the three relations: a is left divisor of b, a is right divisor of b, a divides b, as ordering relations defining a poset . Give for a subset s of the definitions of g. c. (greatest common) left divisor, g. c. right divisor, g. c. divisor of s corresponding to the meet in multiplicative terms. Also, give the definitions of l. c. (least common) left multiple, 1. c. right multiple, l. c. multiple of s corresponding to the join in multiplicative terms.
g) An element is called a unit if it is both a left divisor and a right divisor of 1. The units form a subgroup 
() of .
h) The element n of is called a zero element of if nx = xn = n for every element x of . Show that the divisors of n form a sub-semigroup. Show that there is at most one zero element.
i) The element e is called an idempotent if ee = e and if e is not a zero element. An idempotent is a left divisor (right divisor) of the element a if and only if it is a left unit (right unit) of a.
j) If there is no idempotent other than 1, then each divisor of 1 is a unit and all the divisors of 1 form a normal divisor () of . Moreover, if is commutative, then congruence modulo coincides with equivalence as defined under e) for every pair of non-divisors of zero.
k) If is commutative, then an element is a divisor of 1 if and only if it is a unit.
24. A sub-semigroup of a group is called a halfgroup. Show the following:
a) A finite halfgroup is characterized as a finite semi-group for which the cancellation laws of multiplication are satisfied.
b) An abelian halfgroup is characterized as a semi-group satisfying the commutative and the cancellation laws of multiplication (see § 7, Ex. 4 and also Ex. 25).
c) (O. Ore.) If a semi-group satisfies the cancellation laws of multiplication and also the rule a = a for each a contained in , then it is a halfgroup. (Hint: Form the quotient group of consisting of the formal quotients a/ b(a, b any two elements in ) where a/ b = c /d if there are elements e, f such that ea = fb, ec = fd, and where a/b · c/d = e/f means that there is an element g such that a/b = e/g, c/ d = g /f ; show that the mapping of a onto the quotient aa/a gives an isomorphism of into the quotient group.)
d) (Lambek-Mal’cev.) Any halfgroup satisfies, in addition to the cancellation laws of multiplication, certain polyhedral conditions given by the following construction: A finite system P of v vertices, e edges, and f faces is called an abstract Euclidean polyhedron if 1. every edge is incident with precisely two vertices and with precisely two faces, 2. a vertex is incident with a face if and only if there are precisely two edges incident with both of them, 3. the edges e1, e2, …, en which are incident with a given vertex (face) form a cycle such that with suitable renumbering ei and ei+1 are incident with the same face (vertex), where n is greater than 1 and en+1 = e1, 4. v + f = e+ 2 .The subset of P formed by an edge e and the face F incident with e is called the F-side of e. The subset of P formed by the vertex V incident with the face F is called the angle at V on the F-side of a, b where a, b are the two edges incident with both Pand F. Assign to each angle and to each side an element of such that the relations xa = yb are satisfied, where x, y are assigned to the two sides of an edge e, say to the F-side and to the G-side, and a, b are assigned to the angles formed at a vertex incident with e and F, G respectively. The polyhedral condition corresponding to P states that any one of the finitely many relations xa = yb explained above is a consequence of all the others. (Hint: Apply induction on v; amalgamate adjacent faces.)
e) If a semi-group satisfies the cancellation laws of multiplication and the polyhedral conditions given under d), then it is a halfgroup (see A. Mal’cev, On the embed- ding of associative systems in groups. Mat. Sbornik, Vol. 6 (48) (1939), pp. 331—336; J. Lambek, The immersibility of a semi-group into a group, Canadian Journal of Math., Vol. 3 (1951), pp. 34—43).
f) There are semi-groups satisfying any given finite subsystem of the polyhedral conditions given under d) and also the cancellation laws of multiplication which are not halfgroups (see A. Mal’cev, On the embedding of associative systems in groups II, Mat. Sbornik, Vol. 8 (50) (1940), pp. 251—264).
25. An element d of a multiplicative semi-group is called a denominator if a) dx = xd for any x contained in , b) dx = dy implies x = y.
a) Assuming denominators exist, show that they form an abelian halfgroup d() (see § 7, Ex. 1).
b) Show that the elements of , and the formal quotients a/d where a is contained in and d is contained in d (), together with the symbol 1, form a multiplicative semi-group Q() (called the quotient semi-group of ) by introducing the rules: a = b in Q() if a = b in , a = b/d if ad = b, b/d = a if b = ad, a/d = a′/d′ if ab′ = da′, 1 = a if a is unit element of , a = 1 if 1 = a, d/d = 1, 1 = d/d; ab as in , a · b/d =(ab)/d, b/ d·a = (ba)/d, a/d · a′/ d′ = (aa′)/(dd′), la = a1 = a, 1 · a/ d = a / d·1 = a /d, 1·1 = 1.
c) Show that is a sub-semigroup of Q(). (As regards b) and c), compare § 7, Ex. 2.)
d) Show that the denominators of Q() form an abelian group dQ() with 1 as unit element and that dQ() = Qd()(see § 7, Ex. 3).
e) Show that Q() = if and only if d() is a group (see § 7, Ex. 4).
f) Let 
be a multiplicative semi-group containing a sub-semigroup , where the elements of d() generate a subgroup of whose unit element is the unit element of . Show that there is one and only one homomorphism of Q() into leaving every element of invariant, namely the isomorphism which maps a onto a, a/d onto ad-1, and 1 onto the unity element of .
26. a) Let be an associative semi-ring. Assume that the multiplicative semi-group belonging to contains denominators. Prove that the quotient semi-group of the multiplicative semi-group belonging to forms an associative semi-ring Q() if the addition is defined as follows: a + b as in , a +(b/d) = (ad + b)/d, (b/d) + a = (b + da)/d, (a/d) + (a′/d′) = (ad′ + a′d)/(dd′).
b) The semi-ring Q() is called the quotient semi-ring. Show that it contains as sub-semiring.
c) If is a ring, show that Q() is a ring. Q() is called the quotient ring of .
d) Let be an associative semi-ring containing as sub-semiring, where the elements d() generate a multiplicative group in whose unit element is also the unit element of . Show that there is one and only one homomorphism of the quotient semi-ring Q () into leaving every element of invariant, namely the isomorphism that maps a onto a and a/d onto ad-1.
27. A subring of a field is called a half field. Show that a ring is a half field if and only if the multiplication is commutative and if a product vanishes only if at least one of the factors vanishes. The quotient ring of a half field is a field, which is called the quotient field of the half field. A half field with unit element is called an integral domain or domain of integrity. Give examples.
28. A multiplicative domain is called ordered if there is a binary relation a > b on , called the ordering relation on , such that 1. a is not greater than a, 2. if a > b, b > c, then a > c, 3. if a is neither equal to b nor greater than b, then b > a, 4. if a > b, then ca > cb and ac > bc. Show the following:
a) The ordered multiplicative domain satisfies the cancellation laws of multiplication.
b) From a > b, c > d it follows that ac > bd.
c) The binary relation ‘a ≥ b if b is not greater than a’ is both reflexive and transitive. Furthermore, between any two elements a, b of one of the two relations a ≥ b, b ≥ a holds. If a ≥ b, b ≥ a then a = b. If a ≥ b, c ≥ d then ac ≥ bd.
d) For an ordered group which is not 1, the elements > 1 form a semi-group with the following properties: 1. If a, b are contained in and a is not a left divisor of b then either a = b or b is a left divisor of a;2.a = a for every element a of ; 3. a is not left divisor of a; 4. the cancellation laws of multiplication are satisfied in .
e) A semi-group with the properties 1.-4. mentioned under d) can be embedded into an ordered group so that consists of all the elements > 1. The ordering of the embedding group is uniquely determined (use Ex. 24 c)).
29. A quasi-ring 
having a binary ordering relation is called an ordered quasiring if its elements under addition form an ordered module and if the elements > 0 (called the positive elements) form an ordered semi-group under multiplication. Show that
a) for each element a one, and only one, of the three relations a > 0, a = 0, — a > 0 is true;
b) if the function signa is defined to be 1, 0, —1 according as a < 0, a = 0, — a > 0, respectively, then sign (ab) = sign a · sign b;
c) if the absolute value | a | of a is defined as a or —a according as a ≥ 0 or — a > 0, respectively, then the absolute value also is a multiplicative function: | a · b| = | a | · | b| ; moreover, the triangle inequality |a + b | ≤ | a| + | b | holds;
d) the two inequalities a > b, c > d imply the inequality ac + bd > ad + bc;
e) the ordering of an ordered ring can be extended to an ordering of its quotient ring as follows: a > b/c if ac2 > bc, a/b > c if ab > cb2, a/b > c/d if abd2 > cdb2;
f) that there is only one possible way to extend the ordering of an ordered ring to an ordering of the quotient field.
30. A semi-ring having a binary ordering relation is called ordered if its elements under addition form an ordered semi-module and if, further, from a > b, c > d it follows that ac + bd > bc + ad. Show that
a) the ordering of can be extended to an ordering of the difference ring d() by introducing the rules: a > b — c if a + c > b, a — b > c if a > c + b, a — b > c — d if a + d > c + b, a> 0 if a + a > a, 0 > a if a > a + a, a — b>0 if a > b;
b) there is only one possible way of extending the given ordering of to an ordering of the difference ring.
31. Show that the positive elements of an ordered quasi-ring Q form a sub-semiring P having the property that for any element a 
0 one and only one of the two elements a, — a belongs to P. If, conversely, Q is a quasi-ring with a sub-semiring P that has the algebraic property outlined in the previous statement, then Q is ordered by the ordering relation: a > b if a — b belongs to P.
32. Define recursively the powers of a semi-ring :
and show that they form two-sided ideals satisfying the rule 
. If is associative, then 
If is a Lie-ring, then 
(here the convention regarding the definition of m1m2 as a sub-semimodule if m1,m2 are sub-semimodules is to be used).
33. Show that the subsets of a given set S form a commutative ring B(S) if addition and multiplication of the two subsets a, b of S are defined as follows: a + b is the complement of the intersection of a and b relative to the union of a and b ; a · b is the intersection of a and b. Show that this ring satisfies the laws of a Boolean ring, namely, the laws of a commutative ring and the laws: a a = a, a + a = 0.
34. A mapping a of a multiplicative domain into the multiplicative domain 
is called an anti-homomorphism if a(xy) = a(y)a(x). Show that
a) homomorphisms and anti-homomorphisms combine in a way analogous to that indicated for lattices in § 5;
b) for a group, the mapping that maps each element onto its inverse is an antiautomorphism ;
c) the automorphisms and the anti-automorphisms of a multiplicative domain form a group in which the automorphisms form a normal subgroup of index 1 or 2;
d) every multiplicative domain is anti-isomorphic to its dual domain (–1), which is defined to be the multiplicative domain that arises from if multiplication is redefined by taking as the new product of the factors a and b, in this order, the old product ba;
e) a multiplicative domain is isomorphic with its dual if and only if it has an antiautomorphism ;
f) every group is isomorphic with its dual;
g) the preceding statements remain true for semi-rings if an anti-homomorphism of a semi-ring into a semi-ring is defined to be a mapping a of into satisfying the conditions: a(xy) = a (y)a(x), a(x + y) = a (x) + a(y);
h) every anti-homomorphism a between an associative semi-ring and an associative semi-ring induces the anti-homomorphism between the matrix rings Mn() and Mn() which maps the matrix (βik)) onto the matrix (a(βki));
i) if is a commutative and associative semi-ring, then the correspondence between the matrix (βik)) and its transpose (βki)), which we denote by (βik))T, gives an antiautomorphism of Mn().
35. A derivation of a quasi-ring Q is defined to be a mapping d of Q into Q satisfying the following rules of formal differentiation: d(a + b) = d(a) + d(b),d (ab) = d (a) b+ ad(b). Show that the derivations of Q form a subring D(Q) of the Lie-ring belonging to the operator ring of the additive group of Q. Assign to any element a of L the mapping a of L into L that maps x onto ax, and show that the quasi-ring L is a Lie- ring if and only if the left multiplication a is a derivation mapping a onto 0. Show that the derivations a of a Lie-ring L associated with the elements a of L (inner derivations)form an ideal I (L) of the Lie-ring D(L) of all derivations of L.
36. Let be a ring with unit element. The -module is said to be torsion-free if for each denominator d of the equation dx = 0 implies x = 0. Show that the elements of a torsion free -module and the formal quotients u/d, where u belongs to and d is a denominator of , form a torsion-free module over the quotientring Q() of which contains as submodule, if we define: u = v as in , a = v / d if du = v, v/d = u if du = v, u /d = u′/d′ if ud′ = ud; u + v as in , u + (v/d) = (v/d) + u = (ud + v/d, (u/d) + (u′/d′) = (ud′ + u′d)/dd′, βu as in ; β(ν/d) = (βν)/d, (β/d)u = (βu)/d, (β/ d)(u/d′) = (β u)/dd’.
The Q()-module just defined is called the quotient module of over . Show that the quotient module is generated over Q() by its submodule and thus may be denoted by Q(). Every homomorphism over of into a Q()-module (mn) can be extended in one and only one way to a homomorphism over Q() of the Q ()- module Q() into the Q()-module , viz., the mapping that maps u/d onto d-1h(u).
37. Let be a ring with unit element. An -submodule m of an -module is called primitive in if the -factor module /
is ptorsion free. Show that
a) all elements with torsion in , i. e., the elements of that are annihilated by some denominator of , form a primitive -submodule () of ;
b) the torsion submodule () defined under a) is the smallest primitive -sub - module of ;
c) the intersection of any system of primitive -submodules is a primitive -submodule, and hence for every subset of there is precisely one smallest primitive -submodule ′ containing ;
d) if the subset occurring in c) is an -submodule, then the -factor module ′/ is the torsion module of /.
38. Let be a ring with unit element and let be a torsion-free -quasi-ring in which du = ud for all denominators d of and all elements u of . Show that the quotient module Q () defined in Ex. 36 becomes a Q ()-quasi-ring if the following rules of multiplication are introduced:
(note that we set: uβ as in ,u(β/d) = (uβ)/d, (u/d) β = (u/β)/d, (u/d) (β /d′) = (uβ)/(dd′).
Show that the Q()-quasi-ring just defined contains as an -subring and that there is no other way to extend the rules of operation from to Q() so as to embed the -ring into an Q()-quasi-ring. Thus Q() can rightly be called the Q ()-quotient ring of the -quasi-ring .
39. Let be a commutative ring with unit element and let be an -quasi-ring without torsion. Prove that all linear transformations of the -module that are derivations of the quasi-ring form an -Lie-ring L(, ) without torsion such that Q()L(, ) = L(Q (), Q()).