Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

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INVARIANT SUBGROUPS

67

abstract group. Whenever h> the isomorphism is said to
be multiple. The groups H and Q are called complementary
groups as regards G, and the product of their orders is equal to
the order of G. The fact that two different groups may have
the same complementary groups results directly from the
dihedral and the dicyclic groups.
Let q be any element of Q and let s be any one of the elements
of the corresponding co-set. The order of s must be divisible
by the order m of q, since sm is the lowest power of s that occurs
in H. If m is a power of a prime p then there is an element in
the corresponding co-set whose order is also a power of p,
since the group generated by H and this co-set must involve
a larger subgroup whose order is a power of p than H does.
Hence the theorem: The order of any element of a quotient
group divides the orders of all the elements of the corresponding
co-set, and if this order is a power of a prime number the given
co-set involves an element whose order is a power of the same prime.
As a special case of this theorem it may be observed that
every invariant subgroup of index 2 under any group includes
all the elements of odd order contained in this group. Two
elements which belong to the same co-set as regards an invariant
subgroup are sometimes called equivalent with respect to this
invariant subgroup. They are also said to be congruent with
respect to this invariant subgroup as a modulus. It should
be observed that an invariant subgroup has many of the properties of a modulus in elementary number theory. To a smaller
extent these properties belong to all subgroups, and the terms
equivalent and congruent are sometimes used in connection
with any subgroup.
From the separation of the elements of a group into co-sets
it results directly that every subgroup of index p under any
group includes a p'th, p' p, part of the elements of every other
subgroup of G. We proceed to prove that p' <p whenever
the two distinct subgroups G1, G2 in question are conjugate
under G. Suppose that p= p. It must therefore be possible
to write all the elements of G in the form sls2, where si is any
element of G1 and S2 is any element of G2. Hence all the con