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    Group Theory

    Group Theory : Cayley Tables, Cyclic Groups and Isomorphisms

    ? I have the following cayley tables (which is in modulo 9) determine the order of each element . Prove that G is a cyclic group. ? Let be the symmetric group of degree 3 together with composition of maps. Is G isomorphic to ? Justify your answer. ? Let p be a prime number and G a group of order with identity e

    Groups and Fields

    Find H K in {see attachment}, if H={[1],[8]}, and K = {[1], [4], [10], 13], [16],[19]}.

    Direct Product Groups

    Let G1 and G2 be groups, and let G be a direct product of G1 x G2. Let H = {(x1, x2) element G1 x G2 | x2 = e} and let K = {(x1, x2) element G1 x G2| x1 = e}. (a) Show that H and K are subgroups of G. (b) Show that HK = KH = G (c) Show that H [see attachment] K = {(e,e)}.

    Groups and Subgroups

    Let G sub1 and G sub 2 be groups, with subgroups H sub 1 and H sub 2, respectivetly. Show that {(x sub 1, x sub 2) | H sub 1 is an element of H sub 1, x sub 2 is an element of H sub 2} is a subgroup of the direct product G sub 1 x G sub 2.

    Abelian Groups

    Prove that if G1 and G2 are abelian groups, then the direct product G1 x G2 is abelian.

    Groups: Third Sylow Theorem

    Suppose a simple group G of order 660 is a subgroup of the symmetric group S11 (here S is with 11 as a subscript ) and x = {1,2,3,4,5,6,7,8,9,10,11}( a permutation like 1 goes to 2 and so on, I think ) in G. If P equals the span of x ( or <x> ) determine the permutations which generate NGP ( here G is written as the subscript of

    Proofs in Group Theory: Cayley Table, Subsets and Cosets

    Five Problems: Let G=[FORMULA1], with operation given by multiplication modulo 14. 1) by computing the Cayley table of G, or otherwise, show the G is a group. You may assume that without proof multiplication modulo 14 is associative. 2) Prove that the subset H={1,9,11} is a subgroup of G 3) Compute the left cosets of G a

    Group Theory for Symmetric Groups

    1) Recall that So denotes the symmetric group of degree 6, the group of permutations f the numbers 1 to 6 ... see attached for full questions 1) Recall that denotes the symmetric group of degree 6, the group of permutations f the numbers 1 to 6. let . Thus is a bijection, mapping 1 to 3, 2 to 5, etc. let • C

    Group Theory: Matrix Manipulation and Abelian Group

    1) Let G be the set of 2x2 matrices given by show that G is a group under matrix multiplication. You may assume that matrix multiplication is associative. 2) Let G be a group,... prove that G is Abelian. See attachment.

    Cryptology Square Functions

    Suppose that the ADFGVX square is: _ A D F G V X A F L 1 A 0 2 D J D W 3 G U F C I Y B 4 P G R 5 Q 8 V E V 6 K 7 Z M X X S N H O T 9 and that they keyword is : GERMAN Decipher: AFAXA XAFGF XDDAV DAGGX FGXDD XVVAV VGDDV FADVF ADXXX AXA

    Groups : Symmetry

    Note: G' means derived (commutator) subgroup of G and Sn is symmetric group of degree n Please find G' in each case (a) G is abelian (b) G = Sn

    Groups and elements

    If G is a group of order p^k, where p is a prime and k >=, show that G must have an element of order p.

    Abelian group

    If G is any group, define $:G->G by $(g) = g^-1. Show that G is abelian if an only if $ is a homomorphism.

    Cyclic Groups

    Show that every cyclic group Cn of order n is abelian. (Moreover, show that if G is a group, so is GxG)

    Abelian Groups Examples

    Find examples of the following. Explain your answers. (a) A nonabelian group G and a proper normal subgroup S such that G/S is cyclic.

    Find the mean of the group data

    Using the employment information in the table on Alpha Corporation, find the mean for the grouped data. Years of service Frequency 1-5 5 6-10 20 11-15 25 16-20 10 21-25 5 26-30 3

    Group homomorphisms functions

    Homomorphism Problem 4: Let G, G1, and G2 be groups. Let µ1 : G -> G1 and µ2 : G -> G2 be group homomorphisms. Prove that µ : G -> G1 × G2 defined by : µ (x) = (µ1 (x), µ2 (x)), for all x in G, is a well-defined group homomorphism.