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A particular type of automorphism group which exists only for groups. For a group G, the outer automorphism group is the quotient group Aut(G)/Inn(G), which is the ...

The cyclic group C_(10) is the unique Abelian group of group order 10 (the other order-10 group being the non-Abelian D_5). Examples include the integers modulo 10 under ...

The cyclic group C_9 is one of the two Abelian groups of group order 9 (the other order-9 Abelian group being C_3×C_3; there are no non-Abelian groups of order 9). An example ...

The finite group C_2×C_6 is the finite group of order 12 that is the group direct product of the cyclic group C2 and cyclic group C6. It is one of the two Abelian groups of ...

C_3 is the unique group of group order 3. It is both Abelian and cyclic. Examples include the point groups C_3, C_(3v), and C_(3h) and the integers under addition modulo 3 ...

The finite group C_2×C_2 is one of the two distinct groups of group order 4. The name of this group derives from the fact that it is a group direct product of two C_2 ...

The group C_2×C_2×C_2 is one of the three Abelian groups of order 8 (the other two groups are non-Abelian). An example is the modulo multiplication group M_(24) (which is the ...

C_6 is one of the two groups of group order 6 which, unlike D_3, is Abelian. It is also a cyclic. It is isomorphic to C_2×C_3. Examples include the point groups C_6 and S_6, ...

The trivial group, denoted E or <e>, sometimes also called the identity group, is the unique (up to isomorphism) group containing exactly one element e, the identity element. ...

A cycle of a finite group G is a minimal set of elements {A^0,A^1,...,A^n} such that A^0=A^n=I, where I is the identity element. A diagram of a group showing every cycle in ...