Nilpotent group

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A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with {1}.

In group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.[1]

Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups.

Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series.

Definition[edit]

The definition uses the idea of a central series for a group. The following are equivalent definitions for a nilpotent group G:

  • G has a central series of finite length. That is, a series of normal subgroups
where , or equivalently .
  • G has a lower central series terminating in the trivial subgroup after finitely many steps. That is, a series of normal subgroups
where .
  • G has an upper central series terminating in the whole group after finitely many steps. That is, a series of normal subgroups
where and is the subgroup such that .

For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G; and G is said to be nilpotent of class n. (By definition, the length is n if there are different subgroups in the series, including the trivial subgroup and the whole group.)

Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series. If a group has nilpotency class at most n, then it is sometimes called a nil-n group.

It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class 0, and groups of nilpotency class 1 are exactly the non-trivial abelian groups.[2][3]

Examples[edit]

A portion of the Cayley graph of the discrete Heisenberg group, a well-known nilpotent group.
  • As noted above, every abelian group is nilpotent.[2][4]
  • For a small non-abelian example, consider the quaternion group Q8, which is a smallest non-abelian p-group. It has center {1, −1} of order 2, and its upper central series is {1}, {1, −1}, Q8; so it is nilpotent of class 2.
  • The direct product of two nilpotent groups is nilpotent.[5]
  • All finite p-groups are in fact nilpotent (proof). The maximal class of a group of order pn is n - 1. The 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups.
  • Furthermore, every finite nilpotent group is the direct product of p-groups.[6]
  • The Heisenberg group H is an example of non-abelian,[7] infinite nilpotent group.[8] It has nilpotency class 2 with central series 1, Z(H), H.
  • The multiplicative group of upper unitriangular n x n matrices over any field F is a nilpotent group of nilpotent length n - 1 .
  • The multiplicative group of invertible upper triangular n x n matrices over a field F is not in general nilpotent, but is solvable.
  • Any group G such that G/Z(G) is abelian has nilpotency class 2, with central series {1}, Z(G), G.

Explanation of term[edit]

Nilpotent groups are so called because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group of nilpotence degree and an element , the function defined by (where is the commutator of and ) is nilpotent in the sense that the th iteration of the function is trivial: for all in .

This is not a defining characteristic of nilpotent groups: groups for which is nilpotent of degree (in the sense above) are called -Engel groups,[9] and need not be nilpotent in general. They are proven to be nilpotent if they have finite order, and are conjectured to be nilpotent as long as they are finitely generated.

An abelian group is precisely one for which the adjoint action is not just nilpotent but trivial (a 1-Engel group).

Properties[edit]

Since each successive factor group Zi+1/Zi in the upper central series is abelian, and the series is finite, every nilpotent group is a solvable group with a relatively simple structure.

Every subgroup of a nilpotent group of class n is nilpotent of class at most n;[10] in addition, if f is a homomorphism of a nilpotent group of class n, then the image of f is nilpotent[10] of class at most n.

The following statements are equivalent for finite groups,[11] revealing some useful properties of nilpotency:

  • (a) G is a nilpotent group.
  • (b) If H is a proper subgroup of G, then H is a proper normal subgroup of NG(H) (the normalizer of H in G). This is called the normalizer property and can be phrased simply as "normalizers grow".
  • (c) Every Sylow subgroup of G is normal.
  • (d) G is the direct product of its Sylow subgroups.

Proof: (a)→(b): By induction on |G|. If G is abelian, then for any H, NG(H)=G. If not, if Z(G) is not contained in H, then hZHZ−1h−1=h'H'h−1=H, so H·Z(G) normalizers H. If Z(G) is contained in H,then H/Z(G) is contained in G/Z(G). Note, G/Z(G) is a nilpotent group. Thus, there exists an subgroup of G/Z(G) which normalizers H/Z(G) and H/Z(G) is a proper subgroup of it. Therefore, pullback this subgroup to the subgroup in G and it normalizes H. (This proof is the same argument as for p-groups – the only fact we needed was if G is nilpotent then so is G/Z(G) – so the details are omitted.)

(b)→(c): Let p1,p2,...,ps be the distinct primes dividing its order and let Pi in Sylpi(G),1≤is. Let P=Pi for some i and let N=NG(P). Since P is a normal subgroup of N, P is characteristic in N. Since P char N and N is a normal subgroup of NG(N), we get that P is a normal subgroup of NG(N). This means NG(N) is a subgroup of N and hence NG(N)=N. By (b) we must therefore have N=G, which gives (c).

(c)→(d): Let p1,p2,...,ps be the distinct primes dividing its order and let Pi in Sylpi(G),1≤is. For any t, 1≤ts we show inductively that P1P2Pt is isomorphic to P1×P2×…×Pt. Note first that each Pi is normal in G so P1P2Pt is a subgroup of G. Let H be the product P1P2Pt-1 and let K=Pt,so by induction H is isomorphic to P1×P2×…×Pt-1. In particular,|H|=|P1|·|P2|·…·|Pt-1|. Since |K|=|Pt|, the orders of Hand K are relatively prime. Lagrange's Theorem implies the intersection of H and K is equal to 1. By definition,P1P2Pt=HK, hence HK is isomorphic to H×K which is equal to P1×P2×…×Pt. This completes the induction. Now take t=s to obtain (d).

(d)→(a): Easy to obtain Z(P1×P2×…×Ps) is isomorphic to Z(P1)×…×Z(Ps), then G/Z(G)=(P1/Z(P1))×…×(Ps/Z(Ps)). Thus the hypotheses of (d) also hold for G/Z(G). If Pi≠1 then Z(Pi)≠1, so if G≠1,|G/Z(G)| is less than |G|. By induction,G/Z(G) is nilpotent, so G is nilpotent.

The last statement can be extended to infinite groups: if G is a nilpotent group, then every Sylow subgroup Gp of G is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in G (see torsion subgroup).

Many properties of nilpotent groups are shared by hypercentral groups.

Notes[edit]

  1. ^ Dixon, M. R.; Kirichenko, V. V.; Kurdachenko, L. A.; Otal, J.; Semko, N. N.; Shemetkov, L. A.; Subbotin, I. Ya. (2012). "S. N. Chernikov and the development of infinite group theory". Algebra and Discrete Mathematics. 13 (2): 169–208. 
  2. ^ a b Suprunenko (1976). Matrix Groups. p. 205. 
  3. ^ Tabachnikova & Smith (2000). Topics in Group Theory (Springer Undergraduate Mathematics Series). p. 169. 
  4. ^ Hungerford (1974). Algebra. p. 100. 
  5. ^ Zassenhaus (1999). The theory of groups. p. 143. 
  6. ^ Zassenhaus (1999). Theorem 11. p. 143. 
  7. ^ Haeseler (2002). Automatic Sequences (De Gruyter Expositions in Mathematics, 36). p. 15. 
  8. ^ Palmer (2001). Banach algebras and the general theory of *-algebras. p. 1283. 
  9. ^ For the term, compare Engel's theorem, also on nilpotency.
  10. ^ a b Bechtell (1971), p. 51, Theorem 5.1.3
  11. ^ Isaacs (2008), Thm. 1.26

References[edit]