# Generating $p$-groups The $p$-central series of a group $G$ is the descending sequence of subgroups $$ G= P_{0}(G) \geq \ldots \geq P_{i-1}(G) \geq P_i(G) \geq \ldots \geq $$ where $P_i(G)=[P_{i-1}(G),G]P_{i-1}(G)^p$ for $i\geq 1$. If $P_c(G)=1$ and $c$ is the smallest such integer then $G$ has $p$*-class* $c$. A group with $p$-class $c$ is nilpotent and has nilpotency class at most $c$. Let $G$ be a finite $p$-group with Frattini rank $d$ and class $c$. A group $H$ is a *descendant* of $G$ if $H$ has Frattini rank $d$ and the quotient $H/P_c(H)$ is isomorphic to $G$. A group is an *immediate descendant* of $G$ if it is a descendant of $G$ and has class $c+1$. The $p$-group generation algorithm allows the construction of (immediate) descendants of a $p$-group. For a description of this algorithm, see [[Newman, 1977](../../references.md#cite-newman-pgrp-gen), [O'Brien, 1990](../../references.md#cite-obrien-pgroup)]. `SetVerbose ("GeneratepGroups", 1)` will provide information on the progress of the algorithm. ## `GeneratepGroups(p, d, c : parameters): RngIntElt, RngIntElt, RngIntElt -> [GrpPC], RngIntElt` ```magma Exponent : RngIntElt Default: 0 OrderBound: RngIntElt Default: 0 StepSizes : [RngIntElt] Default: [] All : BoolElt Default: true ``` Generate all $d$-generator $p$-class at most $c$ $p$-groups. All groups constructed satisfy the supplied exponent. Given `OrderBound := n`, all groups constructed have order at most $p^n$. Construct descendants of order $p^{(n + s)}$ of a group of order $p^n$ only for $s$ in `StepSizes`. If `true`, return all groups. Otherwise, return only the capable groups (those which have descendants). ## `Descendants(G : parameters): GrpPC -> [GrpPC], RngIntElt` ## `Descendants(G, c : parameters): GrpPC, RngIntElt -> [GrpPC], RngIntElt` Construct descendants of $G$ having $p$-class at most $c$; if $c$ is not supplied, it is assumed to be one larger than the $p$-class of $G$. This function supports the same variable arguments as `GeneratepGroups`. ## `Example: Generating P Groups (ex-3d4202)` ```magma > G := DihedralGroup(GrpPC, 16); > T := Descendants (G, 8); > #T; 12 > H := T[5]; > H; GrpPC : H of order 128 = 2^7 PC-Relations: H.1^2 = H.7, H.2^2 = H.3 * H.4, H.3^2 = H.4 * H.5, H.4^2 = H.5 * H.6, H.5^2 = H.6 * H.7, H.6^2 = H.7, H.2^H.1 = H.2 * H.3, H.3^H.1 = H.3 * H.4, H.4^H.1 = H.4 * H.5, H.5^H.1 = H.5 * H.6, H.6^H.1 = H.6 * H.7 ``` ## `Example: Generatep Groups (ex-e130f7)` What is the soluble length of a 2-generator group of exponent 4? We construct the 2-generator 2-groups having exponent 4. ```magma > T := GeneratepGroups(2, 2, 10: Exponent := 4); > "The number of 2-generator exponent 4 groups is ", # T; The number of 2-generator exponent 4 groups is 26 ``` What are their soluble lengths? ```magma > for i := 1 to #T do > "Group ", i, " has soluble length ", DerivedLength (T[i]); > end for; Group 1 has soluble length 1 Group 2 has soluble length 2 Group 3 has soluble length 2 Group 4 has soluble length 1 Group 5 has soluble length 2 Group 6 has soluble length 2 Group 7 has soluble length 2 Group 8 has soluble length 2 Group 9 has soluble length 2 Group 10 has soluble length 2 Group 11 has soluble length 2 Group 12 has soluble length 2 Group 13 has soluble length 2 Group 14 has soluble length 2 Group 15 has soluble length 2 Group 16 has soluble length 2 Group 17 has soluble length 2 Group 18 has soluble length 2 Group 19 has soluble length 2 Group 20 has soluble length 2 Group 21 has soluble length 3 Group 22 has soluble length 3 Group 23 has soluble length 3 Group 24 has soluble length 3 Group 25 has soluble length 3 Group 26 has soluble length 3 ``` ## `Example: Is Good (ex-4efc58)` Can we find all 2-generator 3-groups of abundance zero? Such groups have order at most $3^5$. First, we define a function which checks the number of conjugacy classes of a group (to determine abundance). ```magma > IsGoodGroup := function(G, k) > > ncl := # Classes(G); > > O := FactoredOrder(G); > p := O[1][1]; > m := O[1][2]; > n := Floor(m / 2); > e := m - n * 2; > Desired := n * (p^2 - 1) + p^e + k * (p - 1) * (p^2 - 1); > > return (Desired eq ncl); > > end function; ``` Then, we generate the potential candidates and check each. ```magma > a := GeneratepGroups (3, 2, 4 : OrderBound := 5); > #a; 42 > > for i := 1 to #a do > G := a[i]; > if IsGoodGroup(G, 0) then > "Group ", i, " of order ", Order(G), " has abundance 0"; > end if; > end for; Group 1 of order 9 has abundance 0 Group 3 of order 27 has abundance 0 Group 4 of order 27 has abundance 0 Group 11 of order 81 has abundance 0 Group 12 of order 81 has abundance 0 Group 13 of order 81 has abundance 0 Group 14 of order 81 has abundance 0 Group 40 of order 243 has abundance 0 Group 41 of order 243 has abundance 0 Group 42 of order 243 has abundance 0 ``` ## `ClassTwo(p, d : parameters): RngIntElt, RngIntElt -> SeqEnum` ## `ClassTwo(p, d, Step : parameters): RngIntElt, RngIntElt, SeqEnum -> SeqEnum` ## `ClassTwo(p, d, s : parameters): RngIntElt, RngIntElt, RngIntElt -> RngIntElt` ```magma Exponent: RngIntElt Default: 0 ``` Count the $d$-generator $p$-groups of $p$-class 2. If $s$ or $Step$ is supplied, then count only those of order $p^{(d + s)}$ or $p^{(d + m)}$ for $m \in Step$. In the first two invocations, the sequence returns a sequence of length $d \choose 2$, whose $m$-th entry is the number of groups of $p^{(d + m)}$. (Some additional entries may be deduced on the basis of duality.) The last invocation returns the number of groups of $p^{(d + s)}$. For details of the algorithm used see [[Eick and O'Brien, 1999](../../references.md#cite-eo-enum)]. If `Exponent` is `true`, count those groups which have exponent $p$. The directive `SetVerbose ("ClassTwo", 1)` will provide information on the progress of the algorithm. ## `Example: Class Two (ex-6ce143)` Count the number of 3-generator p-class 2 5-groups. ```magma > ClassTwo(5, 3); [ 4, 19, 42, 19, 4, 1 ] ``` For example, the number of 3-generator 5-groups of order $5^6$ and $p$-class 2 is precisely 42. Count the number of 4-generator $p$-class 2 5-groups of order $5^7$. ```magma > ClassTwo(5, 4, 3); 6598 ```