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The functions g and f are said to commute with each other if g ∘ f = f ∘ g. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, | x | + 3 = | x + 3 | only when x ≥ 0. The picture shows another example. The composition of one-to-one (injective) functions is always one ...
Notations expressing that f is a functional square root of g are f = g [1/2] and f = g 1/2 [citation needed] [dubious – discuss], or rather f = g 1/2 (see Iterated function#Fractional_iterates_and_flows,_and_negative_iterates), although this leaves the usual ambiguity with taking the function to that power in the multiplicative sense, just as f ² = f ∘ f can be misinterpreted as x ↦ f(x)².
Any group G is the homomorphic image of some free group F S. Let S be a set of generators of G. The natural map φ: F S → G is an epimorphism, which proves the claim. Equivalently, G is isomorphic to a quotient group of some free group F S. If S can be chosen to be finite here, then G is called finitely generated.
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.
Consider the map f : G → Aut(G), from G to the automorphism group of G defined by f(g) = ϕ g, where ϕ g is the automorphism of G defined by f(g)(h) = ϕ g (h) = ghg −1. The function, f is a group homomorphism, and its kernel is precisely the center of G, and its image is called the inner automorphism group of G, denoted Inn(G).
For any f in L 1 (G), the distance between 0 and the closed convex hull in L 1 (G) of the left translates λ(g)f equals |∫f|. Følner condition. For every finite (or compact) subset F of G there is a measurable subset U of G with finite positive Haar measure such that m(U Δ gU)/m(U) is arbitrarily small for g in F. Leptin's condition.
Negatively curved groups (hyperbolic or CAT(0) groups) are always of type F ∞. [7] Such a group is of type F if and only if it is torsion-free. As an example, cocompact S-arithmetic groups in algebraic groups over number fields are of type F ∞. The Borel–Serre compactification shows that this is also the case for non-cocompact arithmetic ...
Group homomorphisms tend to reduce the orders of elements: if f: G → H is a homomorphism, and a is an element of G of finite order, then ord(f(a)) divides ord(a). If f is injective, then ord(f(a)) = ord(a). This can often be used to prove that there are no homomorphisms or no injective homomorphisms, between two explicitly given groups.