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S1SN, S2SN, S3SN
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See also:S1SN, S2SN, S3SN ,. . . , SNSN, when in it each See also:compound See also:symbol SpSQ is replaced by the single symbol Sr that is See also:equivalent to it, is called the multiplication table of the See also:group. It•indicates directly the result of multiplying together in an assigned sequence any number of operations of the group. In each See also:line (and in each See also:column) of the tableau every operation of the group occurs just once. If the letters in the tableau are regarded as See also:mere symbols, the operation of replacing each symbol in the first line by the symbol which stands under it in the pth line is a permutation performed on the set of N symbols. Thus to the N lines of the tableau there corresponds a set of N permutations performed on the N symbols, which includes the identical permutation that leaves each unchanged. Moreover, if SpSQ=Sr,then the result of carrying out in See also:succession the permutations which correspond to the pth and qth lines gives the permutation which corresponds to the rth line. Hence the set of permutations constitutes a group which is simply isomorphic with the given group. Every group of finite See also:order N can therefore be represented in See also:concrete See also:form as a transitive group of permutations on N symbols. The order of any subgroup or operation of G is necessarily finite. If T1(=S1), T2, ..., T„ are the operations of a subgroup H of G, and if E is any operation of G which is not contained in H, P r o p e r t i e s the set of operations ET1, T 2 IT,,, or EH, are all group distinct from each other and from the operations of H. which If the sets H and EH do not exhaust the operations of G, depend on and if E' is an operation not belonging to them, then the the order. operations of the set E'H are distinct from each other and from those of H and H. This See also:process may be continued till the operations of G are exhausted. The order n of H must therefore be a See also:factor of the order N of G. The ratio N/n is called the See also:index of the subgroup H. By taking for H the cyclical subgroup generated by any operation S of G, it follows that the order of S must be a factor of the order of G. Every operation S is permutable with its own See also:powers. Hence there must be some subgroup H of G of greatest possible order, such that every operation of H is permutable with S. Every operation of H transforms S into itself, and every operation of the set HE trans-forms S into the same operation. Hence, when S is transformed by every operation of G, just N/n distinct operations arise if n is the order of H. These operations, and no others, are conjugate to S within G; they are said to form a set of conjugate operations. The number of operations in every conjugate set is therefore a factor of the order of G. In the same way it may be shown that the number of subgroups which are conjugat ,to a given subgroup is a factor of the order of G. An operation which is permutable with every operation of the group is called a self-conjugate operation. The totality of the self-conjugate operations of a group forms a self-conjugate Abelian subgroup, each of whose operations is permutable with every operation of the group. An Abelian group contains subgroups whose orders are any given factors of the order of the group. In fact, since every subgroup See also:Hof an Abelian group G and the corresponding factor See also:groups G/H are Abelian, this result follows immediately by an See also:induction from the See also:case in which the order contains n See also:prime factors to that in which it contains n-1- i. Fora group which is not Abelian no See also:general 's See also:law can be stated as to the existence or non-existence of a Sylow subgroup whose order is an arbitrarily assigned factor theorem. of the order of the group. In this connexion the most important general result, which is See also:independent of any supposition as to the order of the group, is known as Sylow's theorem, which states that if pa is the highest See also:power of a prime p which divides the order of a group G, then G contains a single conjugate set of subgroups of order pa, the number in the set being of the form i +kp. Sylow's theorem may be extended to show that if pa' is a factor of the order of a group, the number of subgroups of order pa' is of the form i +kp. If, however, pa' is not the highest power of p which divides the order, these groups do not in general form a single conjugate set. The importance of Sylow's theorem in discussing the structure of a group of given order need hardly be insisted on. Thus, as a very See also:simple instance, a group whose order is the product piP2 of two primes (pi <P2) must have a self-conjugate subgroup of order p2, since the order of the group contains no factor, other than unity, of the form i +kp2. The same again is true for a group of order pi2p2, unless p1=2, and p2=3. There is one other numerical See also:property of a group connected with its order which is quite general. If N is the order of G, and n a factor of N, the number of operations of G, whose orders are equal to or are factors of n, is a multiple of n. As already defined, a composite group is a group which contains one or more self-conjugate subgroups, whose orders are greater than unity. If H is a self-conjugate subgroup of G, the factor-group G/H may be either simple or composite. In the Composiformer case G can contain no self-conjugate subgroup K, tlon-See also:series which itself contains H; for if it did K/H would be a self- °fagroup. conjugate subgroup of G/H. When G/H is simple, H is said to be a maximum self-conjugate subgroup of G. Suppose now that G being a given composite group, G, G1, G2, .. , Gn, i is a series of subgroups of G, such that each is a maximum self-conjugate sub-group of the preceding; the last See also:term of the series consisting of the identical operation only. Such a series is called a See also:composition-series of G. In general it is not unique, since a group may have two or more maximum self-conjugate subgroups. A composition-series of a group, however it may be chosen, has the property that the number of terms of which it consists is always the same, while the factor-groups G/Gi, Gi/G2, ... , Gn differ only in the sequence in which they occur. It should be noticed that though a group defines uniquely the set of factor-groups that occur in its composition-series, the set of factor-groups do not conversely in general define a single type of group. When the orders of all the factor-groups are primes the group is said to be soluble. If the series of subgroups G, H, K, ..., L, i is chosen so that each is the greatest self-conjugate subgroup of G contained in the previous one, the series is called a See also:chief composition-series of G. All such series derived from a given group may be shown to consist of the same number of terms, and to give rise to the same set of factor-groups, except as regards sequence. The factor-groups of such a series will not, however, necessarily be simple groups. From any chief composition-series a composition-series may be formed by interpolating between any two terms H and K of the series for which H/K is not a simple group, a number of terms hi, h2, ..., h,; and it may be shown that the factor-groups H/hi, h1/h2, ..., h,/K are all simply isomorphic with each other. A group may be represented as isomorphic with itself by trans-forming all its operations by any one of them. In fact, if SpSQ=Sr, then S-1S,S. S-iSgS =S-1S,S. An isomorphism of the Isomorgroup with itself, established in this way, is called an inner isomorphism. It may be regarded as an operation phism of a carried out on the symbols of the operations, being indeed group with a permutation performed on these symbols. The totality itseif. of these operations clearly constitutes a group isomorphic with the given group, and this group is called the group of inner isomorphisms. A group is simply or multiply isomorphic with its group of inner isomorphisms according as it does not or does contain self-conjugate operations other than identity. It may be possible to establish a See also:correspondence between the operations of a group other than those given by the inner isomorphisms, such that if S' is the operation corresponding to S, then S'pS'q=S', is a consequence of SpSQ=Sr. The substitution on the symbols of the operations of a group resulting from such a correspondence is called an See also:outer isomorphism. The totality of the isomorphisms of both kinds constitutes the group of isomorphisms of the given group, and within this the group of inner isomorphisms is a self-conjugate subgroup. Every set of conjugate operations of a group is necessarily transformed into itself by an inner isomorphism, but two or more sets may be interchanged by an outer isomorphism. A subgroup of a group G, which is transformed into itself by every isomorphism of G, is called a characteristic subgroup. A series of groups G, G1, G2, . . ., I, such that each is a maximum characteristic subgroup of G contained in the preceding, may be shown to have the same invariant properties as the subgroups of a composition series. A group which has no characteristic subgroup must be either a simple group or the See also:direct product of a number of simply isomorphic simple groups. It has been seen that every group of finite order can be represented as a group of permutations performed on a set of symbols whose number is equal to the order of the group. In general such Aermuta- a See also:representation is possible with a smaller number of See also:Lion- symbols. Let H be a subgroup of G, and let the operations groups. of G be divided, in respect of H, into the sets H, S2H, S2H, ... , SmH. If S is any operation of G, the sets SH, SS2H, SS3H, ... , SS,,,H differ from the previous sets only in the sequence in which they occur. In fact, if SS, belong to the set S5H, then since H is a group, the set SSPH is identical with the set S,H. Hence, to each operation S of the group will correspond a permutation per-formed on the symbols of the m sets, and to the product of two operations corresponds the product of the two analogous permutations. The set of permutations, therefore, forms a group isomorphic with the given group. Moreover, the isomorphism is simple unless for one or more operations, other. than identity, the sets all remain unaltered. This can only be the case for S, when every operation conjugate to S belongs to H. In this case H would contain a self-conjugate subgroup, and the isomorphism is multiple. The fact that every group of finite order can be represented, generally in several ways, as a group of permutations, gives See also:special importance to such groups. The number of symbols involved in such a representation is called the degree of the group. In accordance with the general See also:definitions already given, a permutation-group is called transitive or intransitive according as it does or does not contain permutations changing any one of the symbols into any other. It is called imprimitive or See also:primitive according as the symbols can or cannot be arranged in sets, such that every permutation of the group changes the symbols of any one set either among themselves or into the symbols of another set. When a group is imprimitive the number of symbols in each set must clearly be the same. The See also:total number of permutations that can be performed on n symbols is n !, and these necessarily constitute a group. It is known as the symmetric group of degree n, the only rational functions of the symbols which are unaltered by all possible permutations being the symmetric functions. When any permutation is carried out on the product of the n(n-I)/2, See also:differences of the n symbols, it must either remain unaltered or its sign must be changed. Those permutations which leave the product unaltered constitute a group of order n !.(2, which is called the alternating group of degree n; it is a self-conjugate subgroup of the symmetric group. Except when n=4 the alternating group is a simple 'group, A group of degree n, which is not contained in the alternating group, must necessarily have a self-conjugate subgroup of index 2, consisting of those of its permutations which belong to the alternating group. Among the various concrete forms in which a group of finite order can be presented the most important is that of a group of linear substitutions. Such groups have already been referred to in connexion with discontinuous groups. Here the number of distinct substitutions is necessarily finite; and to each operation S of a group G of finite order there will correspond a linear substitution s, viz. 1 x,=cm i_lSi2gi(i,j 2, .. ,m), on a set of m variables, such that if ST = U, then st=u. The linear substitutions s, t, u, . . . then constitute a group g with which G is isomorphic; and whether the isomorphism is simple or multiple g is said to give a " representation " of G as a group of linear substitutions. If all the substitutions of g are transformed by the same substitution on the m variables, the (in general) new group of linear substitutions so constituted is said to be " equivalent " with g as a representation of G; and two representations are called " non-equivalent," or " distinct," when one is not capable of being trans-formed into the other. A group of linear substitutions on m variables is said to he " reducible " when it is possible to choose m' (<m) linear functions of the variables which are transformed among themselves by every substitution of the group. When this cannot be done the group is called " irreducible." It can be shown that a group of linear substitutions, of finite order, is always either irreducible, or such that the variables, when suitably chosen, may be divided into sets, each set being irreducibly transformed among themselves. This being so, it is clear that when the irreducible representations of a group of finite order are known, all representations may be built up. It has been seen at the beginning of this See also:section that every group of finite order N can be presented as a group of permutations (i.e. linear substitutions in a limited sense) on N symbols. This group is obviously reducible; in fact, the sum of the symbols remain unaltered by every substitution of the group. The fundamental theorem in connexion with the representations, as an irreducible group of linear substitutions, of a group of finite order N is the following. If r is the number of different sets of conjugate operations in the group, then, when the group of N permutations is completely reduced, (i.) just r distinct irreducible representations occur: (ii.) each of these occurs a number of times equal to the number of symbols on which it operates: (iii.) these irreducible representations exhaust all the distinct irreducible representations of the group. Among these representations what is called the " identical " representation necessarily occurs, i.e. that in which each operation of the group corresponds to leaving a single symbol unchanged. If these representations are denoted by ri, See also:r2, ... , rr, then any re-presentation of the group as a group of linear substitutions, or in particular as a group of permutations, may be uniquely represented by a symbol Eairi, in the sense that the representation when completely reduced will contain the representation ri just ai times for each suffix i. A representation of a group of finite order as an irreducible group of linear substitutions may be presented in an See also:infinite Group number of equivalent forms. If See also:character- x'i=ZS,ixi(i,j =I, 2, . . . , m), istics. is the linear substitution which, in a given irreducible representation of a group of finite order G, corresponds to the operation S, the See also:determinant S12 ... Additional information and CommentsThere are no comments yet for this article.
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