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

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SIMPLY ISOMORPHIC GROUPS

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EXERCISES
1. If an intransitive group of degree n contains exactly k transitive
constituents, the average number of letters in all its substitutions is n-k.
2. Prove that the group of the square involves an invariant subgroup leading to the four-group as a quotient group and that this invariant
subgroup is its commutator subgroup.
3. All the elements which are common to all the subgroups of a complete set of conjugate subgroups constitute an invariant subgroup.
4. To every invariant subgroup of a quotient group there corresponds
an invariant subgroup of the group, and to every subgroup which involves
a given invariant subgroup there corresponds a subgroup in the quotient
group corresponding to this invariant subgroup.
5. If every element of a group is raised to the same power and if
this power is prime to the order of the group, each element of the group is
found once and only once among these powers.
6. The commutator subgroup of the symmetric group of degree n is
the alternating group of this degree, and the alternating group of degree
n is perfect whenever n>4.
7. A necessary and sufficient condition that the +-subgroup of a cyclic
group is the identity is that the order of this group is not divisible by the
square of a prime number.
30. Simply Isomorphic Groups. One of the most important
and most difficult problems in group theory is to determine
whether two given groups of the same order are simply isomorphic or not. If they are simply isomorphic they are identical
as abstract groups and vice versa. Two cyclic groups of the
same order are always simply isomorphic and a cyclic group
cannot be simply isomorphic with a non-cyclic group. One
of the most useful theorems as regards simply isomorphic groups
may be stated as follows: Two groups of the same order G1,
G2 are simply' isomorphic if they contain two simply isomorphic
invariant subgroups H1, H2 respectively, and are generated by these
subgroups and two elements ti, t2 such that if tic is the lowest power
of ti which occurs in H1, then t2a is the lowest power of t2 that occurs
in H2, and tic, t2a correspond in the given simple isomorphism
of H1, H2. Moreover, it is assumed that tl, t2 transform corresponding generators of H1, H2 into corresponding elements in the
given simple isomorphism.
To illustrate the use of this theorem, let G1, G2 represent two