Prosolvable group
In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently, a group is called prosolvable, if, viewed as a topological group, every open neighborhood of the identity contains a normal subgroup whose corresponding quotient group is a solvable group.
Examples
- Let p be a prime, and denote the field of p-adic numbers, as usual, by . Then the Galois group , where denotes the algebraic closure of , is prosolvable. This follows from the fact that, for any finite Galois extension of , the Galois group can be written as semidirect product , with cyclic of order for some , cyclic of order dividing , and of -power order. Therefore, is solvable.[1]
gollark: What's the point of that?
gollark: Please stop doing that.
gollark: Why are you *deliberately obfuscating your code*?!
gollark: `peripheral.find("modem", rednet.open)` - kind of hacky but works.
gollark: ```luarn = function() sides = {"right","left","top","bottom","front","back"} sides_raw = {"right","left","top","bottom","front","back"} for i=1, #sides do sides[i] = peripheral.getType(sides_raw[i]) if sides[i] == "modem" then rednet.open(sides_raw[i]) print("Rednet Open on "..sides_raw[i].." Side") end endend```WHY
See also
References
- Boston, Nigel (2003), The Proof of Fermat's Last Theorem (PDF), Madison, Wisconsin, USA: University of Wisconsin Press
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