Group isomorphism

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.[1]

Definition and notationEdit

Given two groups   and   a group isomorphism from   to   is a bijective group homomorphism from   to   Spelled out, this means that a group isomorphism is a bijective function   such that for all   and   in   it holds that

 

The two groups   and   are isomorphic if there exists an isomorphism from one to the other.[1][2] This is written

 

Often shorter and simpler notations can be used. When the relevant group operations are understood, they are omitted and one writes

 

Sometimes one can even simply write   Whether such a notation is possible without confusion or ambiguity depends on context. For example, the equals sign is not very suitable when the groups are both subgroups of the same group. See also the examples.

Conversely, given a group   a set   and a bijection   we can make   a group   by defining

 

If   and   then the bijection is an automorphism (q.v.).

Intuitively, group theorists view two isomorphic groups as follows: For every element   of a group   there exists an element   of   such that   "behaves in the same way" as   (operates with other elements of the group in the same way as  ). For instance, if   generates   then so does   This implies, in particular, that   and   are in bijective correspondence. Thus, the definition of an isomorphism is quite natural.

An isomorphism of groups may equivalently be defined as an invertible group homomorphism (the inverse function of a bijective group homomorphism is also a group homomorphism).

ExamplesEdit

In this section some notable examples of isomorphic groups are listed.

  • The group of all real numbers under addition,  , is isomorphic to the group of positive real numbers under multiplication  :
      via the isomorphism  .
  • The group   of integers (with addition) is a subgroup of   and the factor group   is isomorphic to the group   of complex numbers of absolute value 1 (under multiplication):
     
  • The Klein four-group is isomorphic to the direct product of two copies of  , and can therefore be written   Another notation is   because it is a dihedral group.
  • Generalizing this, for all odd     is isomorphic to the direct product of   and  
  • If   is an infinite cyclic group, then   is isomorphic to the integers (with the addition operation). From an algebraic point of view, this means that the set of all integers (with the addition operation) is the "only" infinite cyclic group.

Some groups can be proven to be isomorphic, relying on the axiom of choice, but the proof does not indicate how to construct a concrete isomorphism. Examples:

  • The group   is isomorphic to the group   of all complex numbers under addition.[3]
  • The group   of non-zero complex numbers with multiplication as the operation is isomorphic to the group   mentioned above.

PropertiesEdit

The kernel of an isomorphism from   to   is always {eG}, where eG is the identity of the group  

If   and   are isomorphic, then   is abelian if and only if   is abelian.

If   is an isomorphism from   to   then for any   the order of   equals the order of  

If   and   are isomorphic, then   is a locally finite group if and only if   is locally finite.

The number of distinct groups (up to isomorphism) of order   is given by sequence A000001 in the OEIS. The first few numbers are 0, 1, 1, 1 and 2 meaning that 4 is the lowest order with more than one group.

Cyclic groupsEdit

All cyclic groups of a given order are isomorphic to   where   denotes addition modulo  

Let   be a cyclic group and   be the order of   Letting   be a generator of  ,   is then equal to   We will show that

 

Define

 
so that   Clearly,   is bijective. Then
 
which proves that  

ConsequencesEdit

From the definition, it follows that any isomorphism   will map the identity element of   to the identity element of  

 
that it will map inverses to inverses,
 
and more generally,  th powers to  th powers,
 
and that the inverse map   is also a group isomorphism.

The relation "being isomorphic" satisfies is an equivalence relation. If   is an isomorphism between two groups   and   then everything that is true about   that is only related to the group structure can be translated via   into a true ditto statement about   and vice versa.

AutomorphismsEdit

An isomorphism from a group   to itself is called an automorphism of the group. Thus it is a bijection   such that

 

The image under an automorphism of a conjugacy class is always a conjugacy class (the same or another).

The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group   denoted by   forms itself a group, the automorphism group of  

For all abelian groups there is at least the automorphism that replaces the group elements by their inverses. However, in groups where all elements are equal to their inverses this is the trivial automorphism, e.g. in the Klein four-group. For that group all permutations of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to   (which itself is isomorphic to  ).

In   for a prime number   one non-identity element can be replaced by any other, with corresponding changes in the other elements. The automorphism group is isomorphic to   For example, for   multiplying all elements of   by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because   while lower powers do not give 1. Thus this automorphism generates   There is one more automorphism with this property: multiplying all elements of   by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of   in that order or conversely.

The automorphism group of   is isomorphic to   because only each of the two elements 1 and 5 generate   so apart from the identity we can only interchange these.

The automorphism group of   has order 168, as can be found as follows. All 7 non-identity elements play the same role, so we can choose which plays the role of   Any of the remaining 6 can be chosen to play the role of (0,1,0). This determines which element corresponds to   For   we can choose from 4, which determines the rest. Thus we have   automorphisms. They correspond to those of the Fano plane, of which the 7 points correspond to the 7 non-identity elements. The lines connecting three points correspond to the group operation:   and   on one line means     and   See also general linear group over finite fields.

For abelian groups, all non-trivial automorphisms are outer automorphisms.

Non-abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms.

See alsoEdit

ReferencesEdit

  • Herstein, I. N. (1975). Topics in Algebra (2nd ed.). New York: John Wiley & Sons. ISBN 0471010901.
  1. ^ a b Barnard, Tony & Neil, Hugh (2017). Discovering Group Theory: A Transition to Advanced Mathematics. Boca Ratan: CRC Press. p. 94. ISBN 9781138030169.
  2. ^ Budden, F. J. (1972). The Fascination of Groups (PDF). Cambridge: Cambridge University Press. p. 142. ISBN 0521080169. Retrieved 12 October 2022 – via VDOC.PUB.
  3. ^ Ash (1973). "A Consequence of the Axiom of Choice". Journal of the Australian Mathematical Society. 19 (3): 306–308. doi:10.1017/S1446788700031505. Retrieved 21 September 2013.