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|We may also consider the following subsets of G: • the set Tor(G) of its elements |
of finite order, the torsion of G, • the set G[n] of its elements of order dividing n, for
any n < u>, • the set Gp of its p-elements, the p-part of G, for any prime p, and ...
|As Fp is the union ∪i≥1Fpi of finite fields, any g ∈ G lies in GLn(Fq) for some |
power q of p, so has finite order. Then gs is ... By the uniqueness of the p- and p-
parts of elements of finite groups, the claim follows. See [32, Thm. 15.3] for a
|7}) = r, and part (i) of the lemma is proved. (ii) The proof is analogous to the last |
part of (i) taking S, in place of T,-. Let x be an element of a group G and let p be a
prime. Then x is called a p-element (or “p-singular”) if its order is a power of p, ...
|D The treatment of divisible groups containing nontrivial elements of finite order |
needs some more preparation. ... Thus the union Ap := UneN ker E?-A ls a
subgroup, called the p-part of A, or a primary subgroup of A. 4.19 Example. Using
|Linear algebraic groups, or F-groups, are defined as affine F-varieties with a |
group structure such that multiplication and ... any element of G is of finite order
and its Jordan decomposition coincides with its decomposition into p-part and p' -
|The theory of abstract finite simple groups is built on the intimate relations |
between general group theory, ordinary ... Brauer began his lecture at the
International Congress in 1954 as follows: “The theory of groups of finite order
has been ... It yields an explicit character formula χ(g) for all irreducible
characters χ of G belonging to a p-block B of G with defect group D and all
elements g of G whose p-part gp ...
|This implies that R has a basis as a Z-module consisting of the elements 1, w, a>|
2, a>*~i, where n = deg /(Z). We shall ... Finally two elements x, y of G will be said
to be p-conjugate if their /)'-parts are conjugate in the ordinary sense. Since the ...
|The groups H and Q are called complementary groups as regards G, and the |
product of their orders is equal to the order ... co-sets it results directly that every
subgroup of index p under any group includes a p'th, p' 2 p, part of the elements
|We deduce that the integers ]_[K€c,(G0) |CG(a:K)| and det(C) have the same p-|
part, as desired. I We now discuss ... If G is a group, a p'-section of G is the set of
all elements of G whose p' -part is conjugate to some fixed 9: G G0. This is ...
important p'-section of G is the set G1, of the elements of G which have p-power
|Many authors print the conjugacy classes in various kinds of order that reflects |
their makeup. For instance an element of order p might immediately be followed
by the elements of orders 2p, 3p, . . . that have it for their p-part. This helps the ...