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Indeed, for p-groups, the rank of the group P is the dimension of the vector space P/Φ(P), where Φ(P) is the Frattini subgroup. The rank of a group is also often defined in such a way as to ensure subgroups have rank less than or equal to the whole group, which is automatically the case for dimensions of vector spaces, but not for groups such ...
A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a (nonabelian!) free group of rank at least 2 has subgroups of all countable ranks. The commutator subgroup of a free group of rank k > 1 has infinite rank; for example for F(a,b), it is freely generated by the commutators [a m, b n] for non-zero m and n ...
In fact, these groups are the groups with the largest known minimal non-trivial representation, as a function of group order. The alternating groups ( A n ) {\displaystyle (A_{n})} are quasirandom, since its smallest non-trivial representation has dimension n − 1. {\displaystyle n-1.}
The compact group E 8 is unique among simple compact Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra E 8 itself; it is also the unique one that has the following four properties: trivial center, compact, simply connected, and simply laced (all ...
Other surprising examples include torsion-free rank 2 groups A n,m and B n,m such that A n is isomorphic to B n if and only if n is divisible by m. For abelian groups of infinite rank, there is an example of a group K and a subgroup G such that K is indecomposable; K is generated by G and a single other element; and
A non-finitely generated countable example is given by the additive group of the polynomial ring [] (the free abelian group of countable rank). More complicated examples are the additive group of the rational field Q {\displaystyle \mathbb {Q} } , or its subgroups such as Z [ p − 1 ] {\displaystyle \mathbb {Z} [p^{-1}]} (rational numbers ...
As a Lie group, it is also isomorphic to the 1-torus, which explains the picture of diagonalisable algebraic groups as tori. Any real torus is isogenous to a finite sum of those two; for example the real torus is doubly covered by (but not isomorphic to) . This gives an example of isogenous, non-isomorphic tori.
The trivial group is the only group of order one, and the cyclic group C p is the only group of order p. There are exactly two groups of order p 2, both abelian, namely C p 2 and C p × C p. For example, the cyclic group C 4 and the Klein four-group V 4 which is C 2 × C 2 are both 2-groups of order 4.