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The quotient group is the same idea, although one ends up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects: in the quotient / , the group structure is used to form a natural "regrouping".
Thus S has been embedded into a quotient group of G/N, and since H ⊆ S was an arbitrary countable group, it follows that G/N is SQ-universal. Since every subgroup H of finite index in a group G contains a normal subgroup N also of finite index in G, [10] it easily follows that: If a group G is SQ-universal then so is any finite index subgroup ...
The infinite alternating group , i.e. the group of even finitely supported permutations of the integers, is simple. This group can be written as the increasing union of the finite simple groups A n {\displaystyle A_{n}} with respect to standard embeddings A n → A n + 1 {\displaystyle A_{n}\rightarrow A_{n+1}} .
In linear algebra, a quotient space is a vector space formed by taking a quotient group, where the quotient homomorphism is a linear map. By extension, in abstract algebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotient algebra. However, the use of the term for the more general cases ...
Its Galois group over the base field is the quotient group / = {[], []}, where [g] denotes the coset of g modulo H; that is, its only non-trivial automorphism is the complex conjugation g.
The question of what groups are extensions of by is called the extension problem, and has been studied heavily since the late nineteenth century.As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups {}, where each {+} is an extension of {} by some simple group.
Moreover, if the quotient group / (also called the Frattini quotient of G) has order , then k is the smallest number of generators for G (that is, the smallest cardinality of a generating set for G). In particular a finite p-group is cyclic if and only if its Frattini quotient is cyclic (of order p).
The second homomorphism maps each element i in Z to an element j in the quotient group; that is, j = i mod 2. Here the hook arrow ↪ {\displaystyle \hookrightarrow } indicates that the map 2× from Z to Z is a monomorphism, and the two-headed arrow ↠ {\displaystyle \twoheadrightarrow } indicates an epimorphism (the map mod 2).