68 MA THEMA TICS: G. A. MILLER PROC. N. A. S. MAXIMAL SUBGROUPS OF A GIVEN GROUP BY G. A. MILLER DEPARTMENT OF MATHEMATICS, UNIVERSrTY OF ILLINOIS Communicated December 13, 1940 A maximal subgroup of a given group G is a proper subgroup of G, that is, a subgroup which is neither the identity nor G itself,' and in order to be maximal it cannot be contained in a larger proper subgroup of G. When every proper subgroup of G is maximal then every proper subgroup of G is also minimal, that is, it is of prime order, and vice versa, when every proper subgroup of G is minimal then every proper subgroup of G is also maximal. In each of these two cases G is one of the two possible groups of order p2, p being a prime number, one of the two possible groups of order pq, p and q being distinct prime numbers such that p - 1 is divisible by q, or G is the cyclic group of order pq where p - 1 is not divisible by q and q - 1 is not divisible by p. If the order of a group is divisible by more than two prime numbers, equal or unequal, it is not possible for all of its proper subgroups to be either maximal or minimal since the Sylow subgroups could obviously not then be maximal when all of these prime numbers are distinct. Suppose that G contains one and only one maximal subgroup. This subgroup must then be invariant and it must be generated by every operator of G which does not appear in it since every operator of G appears in at least one maximal subgroup of G. The cyclic group of order pm, p being an arbitrary prime number and m > 1, is the only group which contains one and only one maximal subgroup. The determination of the maximal subgroups of the groups of order pm is clearly equivalent to the determina- tion of its subgroups of index p. In particular, the cross-cut of all such maximal subgroups is the 4-subgroup of the group and every non-cyclic group of order pm contains at least p + 1 maximal subgroups. There is one and only one abelian group of order pm which contains exactly p + 1 maximal subgroups and involves operators of order pm - 1, and when m > 1 there is also one and only one such non-abelian group, except when m = 3 and p = 2. In this special case there are two such non-abelian groups. Let G be an abelian group which contains two and only two maximal subgroups. It was noted above that the order of G is then divisible by at least two distinct prime numbers and it can obviously not be divisible by more than two distinct prime numbers. Moreover, its two Sylow subgroups must be cyclic and it is the direct product of its two Sylow sub- groups. Its +-subgroup is the direct product of the subgroups of prime index in its Sylow subgroups. It was noted above that when these Sylow subgroups are of prime order then every maximal subgroup of G is also a Downloaded by guest on October 2, 2021 VOL. 27, 1941 MA THEMA TICS: G. A. MILLER 69 minimal subgroup of G. In this special case the 4-subgroup of G is the identity, and the +-subgroup of G is always the cross-cut of its two maximal subgroups; this cross-cut is composed of all the operators of G whose orders are not divisible by at least one of the largest powers of the prime numbers which divide the order of G. If any given group G contains two and only two maximal subgroups, each of them is invariant under G because an operator of one of these two subgroups could not transform the other into the one which contains this operator. The index of each of these subgroups under G is a prime number and their cross-cut is an invariant subgroup of G. The corre- sponding quotient group contains two and only two proper subgroups and hence it is the cyclic group of order pq, p and q being distinct prime numr bers. One of the two given maximal subgroups therefore contains a Sylow subgroup whose order is a power of p while the other contains a Sylow subgroup whose order is a power of q. Each of these two Sylow subgroups is invariant under G since the quotient group of G with respect to the given invariant cross-cut contains only one subgroup of order p and also only one subgroup of order q. As the Sylow subgroups of G whose orders are powers of p are conjugate under the given maximal subgroup of index q as well as under G, each of these Sylow subgroups is invariant under an operator whose order is a power of q contained in G. Similar remarks apply to the Sylow subgroups whose orders are powers of q which appear in one of the given maximal subgroups. Moreover, there can be only one such Sylow subgroup in each case since the quotient group with respect to the given cross-cut is the cyclic group of order pq. It therefore results that G contains the direct product of these two Sylow subgroups since they are both invariant and have only the identity in common. As G is assumed to contain two and only two maximal subgroups it has been proved that the abelian groups which are the direct products of two prime power cyclic groups whose orders are powers of distinct prime numbers are the only groups which separately contain two and only two maximal subgroups. It has now been proved that whenever a group contains either exactly one maximal subgroup or exactly two maximal subgroups it must be abelian. When it contains exactly three maximal subgroups it is not necessarily abelian as results from the octic and the quaternion groups. Maximal subgroups constitute an important part in the theory of permuta- tion groups in view of the fact that a necessary and sufficient condition that a transitive permutation group of degree n is primitive is that the subgroup composed of all its permutations which omit a given letter is maximal, and vice versa. In particular, every symmetric group of degree n > 2 is primitive and every such alternating group is also primitive. It is therefore possible to find at least two distinct groups of every degree Downloaded by guest on October 2, 2021 70 MA THEMA TICS: G. A. MILLER PROC. N. A. S. n > 3 which separately contain at least n maximal subgroups and are non- abelian. Suppose that the group G contains exactly three maximal subgroups and that not all of them are invariant under G. They could not form a simple set of conjugates because they would then be transformed under G according to a permutation group of degree 3. This could not be the cyclic group of degree 3 because the three maximal subgroups could not be transformed cyclically by one of them. It could not be the symmetric group of degree 3 because this contains four maximal subgroups. It would not be possible for one of the three maximal subgroups to be in- variant under G and the other two to be conjugate under G since the operators which are not common to these two conjugate subgroups would have to transform the remaining operators of the other among themselves. Hence it results that when G contains exactly three maximal subgroups they are all invariant under G. It was noted above that every abelian group of order 2' which has exactly two independent generators has also exactly three maximal subgroups and that these are the only abelian groups of order 2' which separately have exactly three maximal subgroups. An infinite system of non- abelian groups of order 2' which separately contain three and only three maximal subgroups was also noted above. The group of lowest order which contains three and only three maximal subgroups is the four-group, which also contains three minimal subgroups. While the maximal sub- groups are of considerable interest the minimal subgroups are so very simple that it is somewhat questionable whether a special term should be used for them since the term groups of prime order is sufficiently descriptive. If G is the group of order pm, m > 2, which is obtained by extending the cyclic group of order pm - 1by an operator of order p which is commutative with the operators of its subgroup of index p but with no other operator of this subgroup of order pm - ' then G is non-abelian. When p > 2 it is conformal with the abelian group of type 1, m - 1, and this is also the case when p = 2 and m > 3. When p = 2 and m = 3 it is either the octic group or the quaternion group. In all cases when m > 2 it contains exactly p + 1 maximal subgroups and hence it contains exactly three such sub- groups when p = 2. Each of these subgroups is abelian and when m > 3 two of them are cyclic while the third is non-cyclic. Another infinite system of groups composed of groups which separately have exactly three maximal subgroups is composed of the cyclic groups whose orders are the products of the powers of three distinct prime numbers. When the order of G is not divisible by three distinct prime numbers then this order is of the form 2"' since it must then be generated by two of the three invariant maximal subgroups and the quotient group with respect to their cross-cut is then the four group.