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Homework 9: Group Theory, Assignments of Abstract Algebra

University of Illinois - ChicagoAbstract Algebra

A collection of problems on group theory, covering topics such as surjective homomorphisms, direct products, normal subgroups, and subgroups. Students are asked to prove various properties of groups, show that certain groups are indecomposable, and use theorems like the first and second homomorphism theorem.

Typology: Assignments

2011/2012

Uploaded on 05/18/2012

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HOMEWORK 9
As usual Gand Hare assumed to be groups.
(1) Let G1,G2be groups.
(a) Show that the map
π1:G1×G2→G1
(g1, g2)7→ g1
and
π2:G1×G2→G2
(g1, g2)7→ g2
are surjective homomorphisms.
(b) Show that ker(π1)∼
=G2and ker(π2)∼
=G1.
(c) Conclude that (G1×G2)/G2∼
=G1(really we are not modding out by
G2here, but only a subgroup of the direct product isomorphic to G2).
(2) In this problem we cover a primitive way of showing that certain groups are
idecomposable, i.e. not (isomorphic to) the direct product of two non-trivial
groups.
(a) Show that the direct product of two abelian groups is abelian.
(b) Use the first part of this problem and the classification of groups of
small order to show that the groups Q8, D4, D5, A4and S3are inde-
composable.
(3) In this problem, we show that D6∼
=D3×Z2.
(a) Show that H:= ha2, biis a normal subgroup of D6which is isomorphic
to D3(or S3).
(b) Show that K:= ha3iis a normal subgroup of D6which is isomorphic
to Z2.
(c) Show that H∩K={e}.
(d) Show that G=hH∪Ki(hint what is the only divisor of 12 larger
than 6?).
(e) Conclude from the direct product recognition theorem that D6∼
=D3×Z2.
(4) In this problem we assume that Gis a finite group whose every non-identity
element has order 2.
(a) Show that Gis abelian. (hint For any two elements, a,b ∈Gwe have
that e= (ab)2).
(b) If X:= {x1,...xn}is a minimal generating set for G(i.e., no proper
subset of Xgenerates G) show that
(1, . . . , n)7→ x11,...xnn
defines an isomorphism from
Z2n−→ G.
(c) Conclude that any group whose order is not a power of 2 contains an
element whose order is not 2.
1
pf3

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As usual G and H are assumed to be groups.

(1) Let G 1 , G 2 be groups. (a) Show that the map π 1 : G 1 × G 2 → G 1 (g 1 , g 2 ) 7 → g 1

and

π 2 : G 1 × G 2 → G 2 (g 1 , g 2 ) 7 → g 2 are surjective homomorphisms. (b) Show that ker(π 1 ) ∼= G 2 and ker(π 2 ) ∼= G 1. (c) Conclude that (G 1 × G 2 ) /G 2 ∼= G 1 (really we are not modding out by G 2 here, but only a subgroup of the direct product isomorphic to G 2 ). (2) In this problem we cover a primitive way of showing that certain groups are idecomposable, i.e. not (isomorphic to) the direct product of two non-trivial groups. (a) Show that the direct product of two abelian groups is abelian. (b) Use the first part of this problem and the classification of groups of small order to show that the groups Q 8 , D 4 , D 5 , A 4 and S 3 are inde- composable. (3) In this problem, we show that D 6 ∼= D 3 × Z 2. (a) Show that H := 〈a^2 , b〉 is a normal subgroup of D 6 which is isomorphic to D 3 (or S 3 ). (b) Show that K := 〈a^3 〉 is a normal subgroup of D 6 which is isomorphic to Z 2. (c) Show that H ∩ K = {e}. (d) Show that G = 〈H ∪ K〉 (hint what is the only divisor of 12 larger than 6?). (e) Conclude from the direct product recognition theorem that D 6 ∼= D 3 × Z 2. (4) In this problem we assume that G is a finite group whose every non-identity element has order 2. (a) Show that G is abelian. (hint For any two elements, a, b ∈ G we have that e = (ab)^2 ). (b) If X := {x 1 ,... xn} is a minimal generating set for G (i.e., no proper subset of X generates G) show that ( 1 ,... , n) 7 → x 1 ^1 ,... xnn defines an isomorphism from Z 2 n^ −→ G. (c) Conclude that any group whose order is not a power of 2 contains an element whose order is not 2. 1

(5) Let T be a subset of { 1 ,... , n}. (a) Show that the set of permutations σ ∈ Sn which satisfy σ(T ) = T is a subgroup of Sn which we denote by ST.^1 (b) Let T c^ := { 1 ,... , n}\T (the complelement of T ). Show that ST c^ = ST (i.e. the two groups are equal, not just isomorphic). (hint I am just asking you to show that a bijection fixes a set if and only if it fixes its compliment). (c) Show that ST ∼= Sm × Sn−m where m is the number of elements in T. (hint, consider the subgroup of ST defined as the set of permutations, σ, which satisfy σ(i) = i for every i ∈ T c.) (6) Suppose that H 1 is a normal subgroup of G 1 and H 2 is a normal subgroup of G 2. (a) Show that H 1 × H 2 is a normal subgroup of G 1 × G 2. (b) By using the first homomorphism theorem, show that (G 1 × G 2 ) / (H 1 × H 2 ) ∼= G 1 /H 1 × G 2 /H 2 (7) Suppose that ϕi : Gi → H (i = 1, 2) are homomorphisms of groups. For ϕ := (ϕ 1 , ϕ 2 ) Define G 1 ×ϕ G 2 := {(g 1 , g 2 ) ∈ G 1 × G 2 : ϕ 1 (g 1 ) = ϕ 2 (g 2 )}. (In parts (c)-(e) we assume that ϕ 1 is surjective) (a) Show that G 1 ×ϕ G 2 is a subgroup of G 1 × G 2 (in particular, it’s a group!). (b) Show that if G 1 ×ϕ G 2 = G 1 × G 2 if and only if both ϕi’s are trivial (i.e. send every element to eH ). (c) Show that the map π 2 from the first question in this assignment re- stricts to a surjective homomorpism π 2 | := π 2 |G 1 ×ϕG 2 : G 1 ×ϕ G 2 −→ G 2. (d) Show that ker(π 2 |) ∼= ker(ϕ 1 ) (This problem is not as bad as it seems as long as one keeps themselves organized! It may be helpful to draw a diagram (directed graph) in the shape of a square with the four functions we are considering as directed edges and the four groups we are considering as the vertices). (e) Use Lagrange’s theorem and the first homomorphism theorem to show that |G 1 ×ϕ G 2 | =

|G 1 | |G 2 |

|H|

(f) Let ϕ 1 := sgn : S 3 −→ Z 2 and ϕ 2 := Z 4 −→ Z 2 a mod 4 7 → a mod 2 (we identify { 1 , − 1 } with Z 2 in the obvious way) and define BD 3 := S 3 ×ϕ Z 4. Show that BD 3 is a non abelian group of order 12 which is not isomorphic to D 6 or A 4. (hint one way to show that BD 3 is not isomorphic to D 6 is by showing that BD 3 has an element of order 4,

(^1) For instance if n = 5 and T = { 1 , 3 }, then (13)(24), (245) are in ST , but (12) is not