Group Theory Past Paper 2021 (Held in 2022) - PDF

### **3 (Sem-3/CBCS) MAT HC 2** 2021 (Held in 2022) ### **MATHEMATICS** (Honours) Paper: MAT-HC-3026 (**Group Theory-1**) Full Marks: 80 Time: Three hours The figures in the margin indicate full marks for the questions. 1. Answer the following questions: 1×10=10 (a) Give the condition on *n...

### **3 (Sem-3/CBCS) MAT HC 2** 2021 (Held in 2022) ### **MATHEMATICS** (Honours) Paper: MAT-HC-3026 (**Group Theory-1**) Full Marks: 80 Time: Three hours The figures in the margin indicate full marks for the questions. 1. Answer the following questions: 1×10=10 (a) Give the condition on *n* under which the set {1, 2, 3,..., *n*-1), *n* > 1 is a group under multiplication modulo *n*. (b) Define a binary operation on the set $$IR^n = \{(a_1, a_2,..., a_n): a_1, a_2, ..., a_n \in IR\}$$ for which it is a group. **(c) What is the centre of the dihedral group of order 2*n*?** **(d) Write the generators of the cyclic group Z (the group of integers) under ordinary addition.** **(e) Show by an example that the decomposition of a permutation into a product of 2-cycles is not unique.** **(f) Find the cycles of the permutation :** $$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 5 & 4 & 3 & 1 & 2 \end{pmatrix}$$ **(g) Find the order of the permutation :** $$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 4 & 6 & 5 & 1 & 3 \end{pmatrix}$$ **(h) Let G be the multiplicative group of all non-singular *n* x *n* matrices over *R* and let *R*** be the multiplicative group of all non-zero real numbers. Define a homomorphism from *G* to *R***.** **(i) What do you mean by an isomorphism between two groups ?** **(j) State the second isomorphism theorem.** 2. Answer the following questions: 2×5=10 **(a) Let *G* be a group and *a* \in *G*. Show that (*a*) is a subgroup of *G*.** **(b) If *G* is a finite group, then order of any element of *G* divides the order of *G*. Justify whether this statement is true or false.** **(c) Show that a group of prime order cannot have any non-trivial subgroup. Is it true for a group of finite composite order ?** **(d) Consider the mapping & from the group of real numbers under addition to itself given by $(x) = [x]$, the greatest integer less than or equal to *x*. Examine whether *&* is a homomorphism.** **(e) Let *&* be an isomorphism from a group *G* onto a group *H*. Prove that *&*⁻¹ is also an isomorphism from *H* onto *G*.** - - - **(c) Consider the group *G* = {1, -1} under multiplication. Define *f*: *Z* → *G* by** f(x) = $$ \left\{ \begin{array}{ll} 1, \; if \; n \; is \; even \\ -1, \; if \; n \; is \; odd \end{array} \right.$$ **Show that *f* is a homomorphism from *Z* to *G*.** **(d) Let *f*: *G* → *G*** be a homomorphism. Let *a* \in *G* be such that *o*(a)=*n* and *o*(f(a)) = *m*. Prove that *o*(f(a))/*o*(a), and if *f* is one-one, then *m* = *n*. 3. Answer the following questions: 5×4=20 **(a) Show that a finite group of even order has at least one element of order 2.** ***Or*** **(b) Let *N* be a normal subgroup of a group *G*. Show that *G*/ *N* is abelian if and only if for all *x*, *y* \in *G*, *xyx*⁻¹ *y*⁻¹ \in *N*.** **(c) Show that if a cyclic subgroup *K* of a group *G* is normal in *G*, then every subgroup of *K* is normal in *G*.** ***Or*** **(d) Show that converse of Lagrange's theorem holds in case of finite cyclic groups.** 4. Answer the following questions: 10×4=40 **(a) Let *G* be a group and *x*, *y* \in *G* be such that *xy*² = *y*³ *x* and *yx*² = *x*² *y*. Then show that *x* = *y* = *e*, where *e* is the identity element of *G*.** ***Or*** **(b) Give an example to show that the product of two subgroups of a group is not a subgroup in general. Also show that if *H* and *K* are two subgroups of a group *G*, then *HK* is a subgroup of *G* if and only if *HK* = *KH*.** **(c) Let *G* be a group and *Z*(G) be the centre of *G*. If *G*/ *Z*(G) is cyclic, then show that *G* is abelian.** ***Or*** **(d) State and prove Lagrange's theorem.** **(e) Let *H* and *K* be two normal subgroups of a group *G* such that *H* \subset *K*. Show that *G*/ *K* = *G*/ *H*/ *K*/ *H*.** ***Or*** **(f) Prove Cayley's theorem.**