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Introduction The Simple Groups Other Groups . . Images of word maps in almost simple groups and quasisimple groups . . . . . Matthew Levy Imperial College London, Supervisor: Nikolay Nikolov Groups St Andrews 2013 . . . . . .

  1. Introduction The Simple Groups Other Groups . . Images of word maps in almost simple groups and quasisimple groups . . . . . Matthew Levy Imperial College London, Supervisor: Nikolay Nikolov Groups St Andrews 2013 . . . . . .

  2. Introduction The Simple Groups Other Groups . Outline . . . Introduction 1 Word maps Images of Word Maps . . . The Simple Groups 2 . . . Other Groups 3 Almost Simple Groups Symmetric Groups Quasisimple Groups . . . . . .

  3. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Outline . . . Introduction 1 Word maps Images of Word Maps . . . The Simple Groups 2 . . . Other Groups 3 Almost Simple Groups Symmetric Groups Quasisimple Groups . . . . . .

  4. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Word maps Let w be an element of the free group of rank k and let G be a group. We can define a word map , w : G k → G , by substitution: w : G k − → G ; ( g 1 , ..., g k ) �− → w ( g 1 , ..., g k ) . For example: w ( x ) = x n ; w ( x , y ) = [ x , y ] . We will denote by G w the verbal image of w over G : G w := { w ( g 1 , ..., g k ) : g i ∈ G } . Define the verbal subgroup , w ( G ) = ⟨ G ± 1 w ⟩ . . . . . . .

  5. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Word maps Let w be an element of the free group of rank k and let G be a group. We can define a word map , w : G k → G , by substitution: w : G k − → G ; ( g 1 , ..., g k ) �− → w ( g 1 , ..., g k ) . For example: w ( x ) = x n ; w ( x , y ) = [ x , y ] . We will denote by G w the verbal image of w over G : G w := { w ( g 1 , ..., g k ) : g i ∈ G } . Define the verbal subgroup , w ( G ) = ⟨ G ± 1 w ⟩ . . . . . . .

  6. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Word maps Let w be an element of the free group of rank k and let G be a group. We can define a word map , w : G k → G , by substitution: w : G k − → G ; ( g 1 , ..., g k ) �− → w ( g 1 , ..., g k ) . For example: w ( x ) = x n ; w ( x , y ) = [ x , y ] . We will denote by G w the verbal image of w over G : G w := { w ( g 1 , ..., g k ) : g i ∈ G } . Define the verbal subgroup , w ( G ) = ⟨ G ± 1 w ⟩ . . . . . . .

  7. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Outline . . . Introduction 1 Word maps Images of Word Maps . . . The Simple Groups 2 . . . Other Groups 3 Almost Simple Groups Symmetric Groups Quasisimple Groups . . . . . .

  8. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Images of Word Maps . Theorem (M. Kassabov & N. Nikolov, Dec 2011) . . . For every n ≥ 7 , n ̸ = 13 there is a word w ( x 1 , x 2 ) ∈ F 2 such that Alt(n ) w consists of the identity and all 3 -cycles. When n = 13 there is word w ( x 1 , x 2 , x 3 ) ∈ F 3 with the same property. . . . . . . . . . . .

  9. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Images of Word Maps . Theorem (M. Kassabov & N. Nikolov, Dec 2011) . . . For every n ≥ 7 , n ̸ = 13 there is a word w ( x 1 , x 2 ) ∈ F 2 such that Alt(n ) w consists of the identity and all 3 -cycles. When n = 13 there is word w ( x 1 , x 2 , x 3 ) ∈ F 3 with the same property. . . . . . Clearly holds for Alt(5), e.g. w ( x ) = x 10 . Also holds for Sym( n ). They go on to give other explicit examples e.g. all p -cycles with p prime 3 < p < n . Similar results for SL( n , q ). More examples can be found in [L.]. They were motivated by verbal width . . . . . . .

  10. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Images of Word Maps . Theorem (M. Kassabov & N. Nikolov, Dec 2011) . . . For every n ≥ 7 , n ̸ = 13 there is a word w ( x 1 , x 2 ) ∈ F 2 such that Alt(n ) w consists of the identity and all 3 -cycles. When n = 13 there is word w ( x 1 , x 2 , x 3 ) ∈ F 3 with the same property. . . . . . Clearly holds for Alt(5), e.g. w ( x ) = x 10 . Also holds for Sym( n ). They go on to give other explicit examples e.g. all p -cycles with p prime 3 < p < n . Similar results for SL( n , q ). More examples can be found in [L.]. They were motivated by verbal width . . . . . . .

  11. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Verbal Width Say w has finite width in G if there exists m such that = { g 1 ... g m : g i ∈ G ± 1 w ( G ) = G ∗ m w } . w Otherwise we say w has infinite width . Define the width to be the least such m . Clearly, if G is finite we always have finite width bounded by | G | . . Theorem (Larsen, Shalev, Tiep) . . . For any w ̸ = 1 we have G = G w G w when G is a sufficiently large finite simple group. . . . . . The requirement of the size of G cannot be removed. . Corollary (Kassabov & Nikolov) . . . For any k, there exists a word w and a finite simple group G, such that w is not an identity in G, but G ̸ = G ∗ k w . . . . . . . . . . . .

  12. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Verbal Width Say w has finite width in G if there exists m such that = { g 1 ... g m : g i ∈ G ± 1 w ( G ) = G ∗ m w } . w Otherwise we say w has infinite width . Define the width to be the least such m . Clearly, if G is finite we always have finite width bounded by | G | . . Theorem (Larsen, Shalev, Tiep) . . . For any w ̸ = 1 we have G = G w G w when G is a sufficiently large finite simple group. . . . . . The requirement of the size of G cannot be removed. . Corollary (Kassabov & Nikolov) . . . For any k, there exists a word w and a finite simple group G, such that w is not an identity in G, but G ̸ = G ∗ k w . . . . . . . . . . . .

  13. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Verbal Width Say w has finite width in G if there exists m such that = { g 1 ... g m : g i ∈ G ± 1 w ( G ) = G ∗ m w } . w Otherwise we say w has infinite width . Define the width to be the least such m . Clearly, if G is finite we always have finite width bounded by | G | . . Theorem (Larsen, Shalev, Tiep) . . . For any w ̸ = 1 we have G = G w G w when G is a sufficiently large finite simple group. . . . . . The requirement of the size of G cannot be removed. . Corollary (Kassabov & Nikolov) . . . For any k, there exists a word w and a finite simple group G, such that w is not an identity in G, but G ̸ = G ∗ k w . . . . . . . . . . . .

  14. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Verbal Width Say w has finite width in G if there exists m such that = { g 1 ... g m : g i ∈ G ± 1 w ( G ) = G ∗ m w } . w Otherwise we say w has infinite width . Define the width to be the least such m . Clearly, if G is finite we always have finite width bounded by | G | . . Theorem (Larsen, Shalev, Tiep) . . . For any w ̸ = 1 we have G = G w G w when G is a sufficiently large finite simple group. . . . . . The requirement of the size of G cannot be removed. . Corollary (Kassabov & Nikolov) . . . For any k, there exists a word w and a finite simple group G, such that w is not an identity in G, but G ̸ = G ∗ k w . . . . . . . . . . . .

  15. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Question Fix a group G and let A be a subset of G . Does there exist a word w such that G w = A ? i.e. What are the verbal images of G ? Two necessary conditions: Clearly we must have e ∈ A (since w ( e , ..., e ) = e ). For every α ∈ Aut( G ), α ( A ) = A (since α ( w ( g 1 , ..., g k )) = w ( α ( g 1 ) , ..., α ( g k )) . If we assume G is a simple group, are these conditions sufficient? . . . . . .

  16. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Question Fix a group G and let A be a subset of G . Does there exist a word w such that G w = A ? i.e. What are the verbal images of G ? Two necessary conditions: Clearly we must have e ∈ A (since w ( e , ..., e ) = e ). For every α ∈ Aut( G ), α ( A ) = A (since α ( w ( g 1 , ..., g k )) = w ( α ( g 1 ) , ..., α ( g k )) . If we assume G is a simple group, are these conditions sufficient? . . . . . .

  17. Introduction Word maps The Simple Groups Images of Word Maps Other Groups . Question Fix a group G and let A be a subset of G . Does there exist a word w such that G w = A ? i.e. What are the verbal images of G ? Two necessary conditions: Clearly we must have e ∈ A (since w ( e , ..., e ) = e ). For every α ∈ Aut( G ), α ( A ) = A (since α ( w ( g 1 , ..., g k )) = w ( α ( g 1 ) , ..., α ( g k )) . If we assume G is a simple group, are these conditions sufficient? . . . . . .

  18. Introduction The Simple Groups Other Groups . The Simple Groups . Theorem (Lubotzky, June 2012) . . . Let G be a finite simple group and A a subset of G such that e ∈ A and for every α ∈ Aut(G), α ( A ) = A. Then there exists a word w ∈ F 2 such that G w = A. . . . . . . . . . . .

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