Normal Complement Problem for Metacyclic Groups Surinder Kaur - - PowerPoint PPT Presentation

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Normal Complement Problem for Metacyclic Groups

Surinder Kaur (Joint work with Manju Khan)

Indian Institute of Technology Ropar email: surinder.kaur@iitrpr.ac.in

Groups, Rings and Associated Structures

June 14, 2019

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Notations

G - a group

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Notations

G - a group F - a field

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Notations

G - a group F - a field FG - group algebra of G over F

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Notations

G - a group F - a field FG - group algebra of G over F U(FG) - unit group of FG

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Notations

G - a group F - a field FG - group algebra of G over F U(FG) - unit group of FG V (FG) - normalized unit group of FG

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Notations

G - a group F - a field FG - group algebra of G over F U(FG) - unit group of FG V (FG) - normalized unit group of FG Γ(H) - two-sided ideal generated by elements (h − 1) in FG

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Normal Complement Problem

Normal Complement Problem (R. K. Dennis* (1977)) For what groups G and rings R there exists a homomorphism φ : U(RG) → G such that φ is split by the natural inclusion i : G → U(RG)?

*R. Keith Dennis, The structure of the unit group of group rings, Lecture Notes in Pure and Appl. Math.,

• Vol. 26, 1977

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Results known by far

Let Lp denote the set of finite p-groups G which have a normal complement in V (FpG). Then

* L. E. Moran and R. N. Tench, Normal complements in mod p-envelopes. Israel J. Math., Israel J. Math.

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Results known by far

Let Lp denote the set of finite p-groups G which have a normal complement in V (FpG). Then Moran and Trench (1977)*

1

Finite ableian p-groups lie in Lp.

* L. E. Moran and R. N. Tench, Normal complements in mod p-envelopes. Israel J. Math., Israel J. Math.

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Results known by far

Let Lp denote the set of finite p-groups G which have a normal complement in V (FpG). Then Moran and Trench (1977)*

1

Finite ableian p-groups lie in Lp.

2

D8 and Q8 belong to L2.

* L. E. Moran and R. N. Tench, Normal complements in mod p-envelopes. Israel J. Math., Israel J. Math.

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Results known by far

Let Lp denote the set of finite p-groups G which have a normal complement in V (FpG). Then Moran and Trench (1977)*

1

Finite ableian p-groups lie in Lp.

2

D8 and Q8 belong to L2.

3

Finite p-groups of exponent p and nilpotency class 2 lie in Lp.

* L. E. Moran and R. N. Tench, Normal complements in mod p-envelopes. Israel J. Math., Israel J. Math.

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Results known by far

Let Lp denote the set of finite p-groups G which have a normal complement in V (FpG). Then Moran and Trench (1977)*

1

Finite ableian p-groups lie in Lp.

2

D8 and Q8 belong to L2.

3

Finite p-groups of exponent p and nilpotency class 2 lie in Lp.

4

D16 / ∈ L2.

* L. E. Moran and R. N. Tench, Normal complements in mod p-envelopes. Israel J. Math., Israel J. Math.

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• L. Ivory (1980)**

1

The dihedral, semi-dihedral and quaternion groups of order 16 and greater do not lie in L2.

**Ivory, Lee R., A note on normal complements in mod p envelopes, Proc. Amer. Math. Soc.

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• L. Ivory (1980)**

1

The dihedral, semi-dihedral and quaternion groups of order 16 and greater do not lie in L2.

2

11 of the 14 groups of order 16 are in L2.

**Ivory, Lee R., A note on normal complements in mod p envelopes, Proc. Amer. Math. Soc.

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Normal complement problem for some metacyclic p-groups

Theorem* (Kaur and Khan) Let G be a finite split metacyclic p-group with nilpotency class 2 and F be the field with p elements. Then G has a normal complement in V (FG).

*S. Kaur, M. Khan, A note on Normal Complement Problem for Split Metacyclic Groups. Communications in

Algebra (2019)

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Sketch of proof: Let G = A ⋊ B, where A = a and B = b are cyclic groups of order pn and pm respectively. Then V (FG) = (1 + Γ(A)) ⋊ V (FB).

*R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra (1984).

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Sketch of proof: Let G = A ⋊ B, where A = a and B = b are cyclic groups of order pn and pm respectively. Then V (FG) = (1 + Γ(A)) ⋊ V (FB). It is known* that V (FB) =

pm−1

• i=1

i=rp

(1 + (b − 1)i). Write V (FB) = M × b.

*R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra (1984).

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Sketch of proof: Let G = A ⋊ B, where A = a and B = b are cyclic groups of order pn and pm respectively. Then V (FG) = (1 + Γ(A)) ⋊ V (FB). It is known* that V (FB) =

pm−1

• i=1

i=rp

(1 + (b − 1)i). Write V (FB) = M × b. Let x = 1 +

pn−1

• i=1

αai (ai − 1) ∈ (1 + Γ(A)), where αai ∈ FB. Define φ : (1 + Γ(A)) → A by the rule x → a

pn−1

i=1

iǫ(αai ).

It can be checked that φ is an epimorphism.

*R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra (1984).

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Sketch of proof: Let G = A ⋊ B, where A = a and B = b are cyclic groups of order pn and pm respectively. Then V (FG) = (1 + Γ(A)) ⋊ V (FB). It is known* that V (FB) =

pm−1

• i=1

i=rp

(1 + (b − 1)i). Write V (FB) = M × b. Let x = 1 +

pn−1

• i=1

αai (ai − 1) ∈ (1 + Γ(A)), where αai ∈ FB. Define φ : (1 + Γ(A)) → A by the rule x → a

pn−1

i=1

iǫ(αai ).

It can be checked that φ is an epimorphism. Thus (1 + Γ(A)) = ker(φ) ⋊ A.

*R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra (1984).

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Sketch of proof: Let G = A ⋊ B, where A = a and B = b are cyclic groups of order pn and pm respectively. Then V (FG) = (1 + Γ(A)) ⋊ V (FB). It is known* that V (FB) =

pm−1

• i=1

i=rp

(1 + (b − 1)i). Write V (FB) = M × b. Let x = 1 +

pn−1

• i=1

αai (ai − 1) ∈ (1 + Γ(A)), where αai ∈ FB. Define φ : (1 + Γ(A)) → A by the rule x → a

pn−1

i=1

iǫ(αai ).

It can be checked that φ is an epimorphism. Thus (1 + Γ(A)) = ker(φ) ⋊ A. Finally, we obtain that V (FG) = (ker(φ) ⋊ M) ⋊ (a ⋊ b).

*R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra (1984).

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Normal complement of D2.3m in V (FD2.3m)

Let p be an odd prime and F be a finite field of characteristic p. Let D2pm = a, b | apm = b2 = e, b−1ab = a−1 be the dihedral group of order 2pm. If A is the normal subgroup of D2pm, then V (FD2pm) = (1 + Γ(A)) ⋊ V (F(b)).

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Normal complement of D2.3m in V (FD2.3m)

Let p be an odd prime and F be a finite field of characteristic p. Let D2pm = a, b | apm = b2 = e, b−1ab = a−1 be the dihedral group of order 2pm. If A is the normal subgroup of D2pm, then V (FD2pm) = (1 + Γ(A)) ⋊ V (F(b)). Theorem Let F be the field with 3 elements. Then the dihedral group D2.3m has a normal complement in the normalized unit group V (FD2.3m).

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Proof: Consider the epimorphism φ : (1 + Γ(A)) → A defined by the rule 1 +

pm−1

• i=1

(αi(ai − 1) + βi(ai − 1)b) → a

pm−1

i=1

i(αi +βi ).

Thus (1 + Γ(A)) = ker(φ) ⋊ A. Since b normalizes ker(φ), we have V (FD2.3m) = ker(φ) ⋊ D2.3m.

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Proof: Consider the epimorphism φ : (1 + Γ(A)) → A defined by the rule 1 +

pm−1

• i=1

(αi(ai − 1) + βi(ai − 1)b) → a

pm−1

i=1

i(αi +βi ).

Thus (1 + Γ(A)) = ker(φ) ⋊ A. Since b normalizes ker(φ), we have V (FD2.3m) = ker(φ) ⋊ D2.3m. Corollary Let FS3 be the group algebra of symmetric group S3 over the field F with 3

• elements. Then the elementary abelian 3-group

N = (1 + (a − 1) + 2(a − 1)b) × Z(V (FS3)) is a normal complement of S3 in V (FS3).

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A Partition of V (FD2pm) \ (1 + Γ(A))

Let V (F(b)) = γ, where γ = α + βb, for some α, β ∈ F. Define the set S =

• γiz | 1 ≤ i ≤ |F ∗|, z ∈ C(1+Γ(A))(b)
• .

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A Partition of V (FD2pm) \ (1 + Γ(A))

Let V (F(b)) = γ, where γ = α + βb, for some α, β ∈ F. Define the set S =

• γiz | 1 ≤ i ≤ |F ∗|, z ∈ C(1+Γ(A))(b)
• .

Theorem V (FD2pm) \ (1 + Γ(A)) =

• s∈S

Cl(s), where Cl(s) denotes the conjugacy class of s in V (FD2pm).

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Normal complement problem for FD2pm, when |F| > 3

Theorem Let F be a finite field of characteristic p, where p is an odd prime. Then D2pm does not have normal complement in V (FD2pm), for |F| > 3.

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Sketch of Proof: Suppose there exists an epimorphism φ : V (FD2pm) → D2pm fixing elements of D2pm such that V (FD2pm) = N ⋊ D2pm, where N = ker(φ).

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Sketch of Proof: Suppose there exists an epimorphism φ : V (FD2pm) → D2pm fixing elements of D2pm such that V (FD2pm) = N ⋊ D2pm, where N = ker(φ). Let I =

• i | 1 ≤ i ≤ |F ∗| and φ(γi) = e
• . Since C(1+Γ(A))(b) ⊆ N, for any

i ∈ I, we have γiz ∈ N.

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Sketch of Proof: Suppose there exists an epimorphism φ : V (FD2pm) → D2pm fixing elements of D2pm such that V (FD2pm) = N ⋊ D2pm, where N = ker(φ). Let I =

• i | 1 ≤ i ≤ |F ∗| and φ(γi) = e
• . Since C(1+Γ(A))(b) ⊆ N, for any

i ∈ I, we have γiz ∈ N. There are |F|(pm−1)(|F| − 3) 2 class representatives in S which are in N \ (1 + Γ(A)).

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Sketch of Proof: Suppose there exists an epimorphism φ : V (FD2pm) → D2pm fixing elements of D2pm such that V (FD2pm) = N ⋊ D2pm, where N = ker(φ). Let I =

• i | 1 ≤ i ≤ |F ∗| and φ(γi) = e
• . Since C(1+Γ(A))(b) ⊆ N, for any

i ∈ I, we have γiz ∈ N. There are |F|(pm−1)(|F| − 3) 2 class representatives in S which are in N \ (1 + Γ(A)). Hence |N \ 1 + Γ(A)| ≥ |F|2(pm−1)(|F| − 3) 2 .

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Sketch of Proof: Suppose there exists an epimorphism φ : V (FD2pm) → D2pm fixing elements of D2pm such that V (FD2pm) = N ⋊ D2pm, where N = ker(φ). Let I =

• i | 1 ≤ i ≤ |F ∗| and φ(γi) = e
• . Since C(1+Γ(A))(b) ⊆ N, for any

i ∈ I, we have γiz ∈ N. There are |F|(pm−1)(|F| − 3) 2 class representatives in S which are in N \ (1 + Γ(A)). Hence |N \ 1 + Γ(A)| ≥ |F|2(pm−1)(|F| − 3) 2 . But |N \ 1 + Γ(A)| = |N| − |N ∩ 1 + Γ(A)| = |F|2(pm−1)(|F| − 3) 2pm .

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Normal complement problem for FD2pm

Theorem Let F be a finite field of characteristic p, where p is an odd prime. Then D2pm has a normal complement in V (FD2pm) if and only if p = 3 and |F| = 3.

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Normal complement problem for FD2pm

Theorem Let F be a finite field of characteristic p, where p is an odd prime. Then D2pm has a normal complement in V (FD2pm) if and only if p = 3 and |F| = 3. Since S4 ∼ = V4 ⋊ S3, we have that

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Normal complement problem for FD2pm

Theorem Let F be a finite field of characteristic p, where p is an odd prime. Then D2pm has a normal complement in V (FD2pm) if and only if p = 3 and |F| = 3. Since S4 ∼ = V4 ⋊ S3, we have that Corollary If F is a finite field of characteristic 3 such that |F| > 3, then S4 does not have a normal complement in V (FS4).

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Concluding Remarks

1

What will be the answer to normal complement problem when G is a finite metacyclic p-group of nilpotency class 3?

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Concluding Remarks

1

What will be the answer to normal complement problem when G is a finite metacyclic p-group of nilpotency class 3?

2

What can be said about the existence of a normal complement to the metacyclic group G = Cp ⋊ Cq, where q | p − 1, in the modular group algebra FG?, for finite field F of characteristic p.

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References

Robert Sandling. Units in the modular group algebra of a finite abelian p-group.

• J. Pure Appl. Algebra, 33(3):337–346, 1984.
• L. E. Moran and R. N. Tench.

Normal complements in mod p-envelopes. Israel J. Math., 27(3-4):331–338, 1977.

• S. Kaur and M. Khan,

Normal Complement Problem and Structure of Unitary Units (preprint). Adalbert Bovdi. The group of units of a group algebra of characteristic p.

• Publ. Math. Debrecen, 52(1-2):193–244, 1998.

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Refernces

• D. L. Johnson.

The modular group-ring of a finite p-group.

• Proc. Amer. Math. Soc., 68(1):19–22, 1978.
• R. Keith Dennis.

The structure of the unit group of group rings. pages 103–130. Lecture Notes in Pure and Appl. Math., Vol. 26, 1977.

• S. K. Sehgal.

Units in integral group rings, volume 69 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. With an appendix by Al Weiss. Lee R. Ivory. A note on normal complements in mod p envelopes.

• Proc. Amer. Math. Soc., 79(1):9–12, 1980.

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References

• K. Kaur, M. Khan, and T. Chatterjee.

A note on normal complement problem.

• J. Algebra Appl., 16(1):1750011, 11, 2017.
• R. K. Sharma, J. B. Srivastava, and Manju Khan.

The unit group of FS3. Acta Math. Acad. Paedagog. Nyh´

• azi. (N.S.), 23(2):129–142, 2007.

Frank R¨

• hl.

Unit groups of completed modular group algebras and the isomorphism problem.

• Proc. Amer. Math. Soc., 111(3):611–618, 1991.

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