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Land plants all have heteromorphic (anisomorphic) alternation of generations, in which the sporophyte and gametophyte are distinctly different. All bryophytes, i.e. liverworts, mosses and hornworts, have the gametophyte generation as the most conspicuous. As an illustration, consider a monoicous moss.
Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, shape'. The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or ...
The study of reproduction and development in organisms was carried out by many botanists and zoologists.. Wilhelm Hofmeister demonstrated that alternation of generations is a feature that unites plants, and published this result in 1851 (see plant sexuality).
An isomorph is an organism that does not change in shape during growth. The implication is that its volume is proportional to its cubed length, and its surface area to its squared length.
A category C consists of two classes, one of objects and the other of morphisms.There are two objects that are associated to every morphism, the source and the target.A morphism f from X to Y is a morphism with source X and target Y; it is commonly written as f : X → Y or X Y the latter form being better suited for commutative diagrams.
Consider a finite group G, a field k and the group algebra kG.The category of k-linear group representations of G is isomorphic to the category of left modules over kG.The isomorphism can be described as follows: given a group representation ρ : G → GL(V), where V is a vector space over k, GL(V) is the group of its k-linear automorphisms, and ρ is a group homomorphism, we turn V into a ...
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 Z p {\displaystyle \mathbb {Z} _{p}} for a prime number p , {\displaystyle p,} one non-identity element can be replaced by any other, with corresponding changes in the other elements.
Spore diversity suggests that there were many plant groups, of which no other remains are known. Some early plants may have had heteromorphic alternation of generations, with later acquisition of isomorphic gametophytes in certain lineages. [30]