The question is Let G and G\'\' be groups (with a,b,c in G and a\',b\',c\' in G\

Question

The question is
Let G and G'' be groups (with a,b,c in G and a',b',c' in G'') where* denotes the operation of both groups. Define * on G X G''(G cross G'') so that (a, a') * (b, b') = (a*b, a'*b'). Briefly explain why GxG'' with this * is a group.

I know the three axioms for a group (associativity, identity andinverse), and I sort of understand the concept of a Cartesianproduct. But I am just completely stuck. Can anyonegive me a hint or a line of thinking that would help me start insolving this problem?

Explanation / Answer

The question is Let G and G'' be groups (with a,b,c in G and a',b',c' in G'') where* denotes the operation of both groups. Define * on G X G''(G cross G'') so that (a, a') * (b, b') = (a*b, a'*b'). Briefly explain why GxG'' with this * is a group. G=[A,B,C].....................G'=[A',B',C'] IF 3 ELEMENTS ARE IN ONE GROUP .THEN ONE OF THEM IS I...... SAY A=I WITHOUT ANY LOSS OF GENERALITY. HENCE A*B=B...A*C=C.. THEN INVERSE OF B IS C AND SO B*C=C*B=A SO WE HAVE B*B=C AND C*C=B SO THE COMPOSITION TABLE FOR G IS    A   B   C A   A   B   C B   B   C   A C   C   A   B SIMILAR TABLE APPLIES FOR G'    A'    B'    C' A'    A'    B'    C' B'    B'    C'    A' C'    C'    A'    B' GXG'=[(A,A'),(A,B'),(A,C'),(B,A'),(B,B'),(B,C'),(C,A'),(C,B'),(C,C')] HERE THERE IS NO SIMPLE WAY TO GO BY WE HAVE TO MAKE THE COMPOSITION TABLE AND PROVE THE 3 RULES YOUMENTIONED. USING THE ABOVE COMPOSITION TABLES FOR G AND G' HERE WE FIND THAT A,A' IS THE IDENTITY ELEMENT SINCE A AND A' AREIDENTITY ELEMENTS AND (A,A')X(Z,Z')=[(A*Z),(A'*Z')]=[Z,Z'] IN GENERAL ....Z IS COMMONREPRESENTATION FOR B OR C. MAKE A TABLE AS GIVEN BELOW AND YOU WILL FIND ALL YOUR REQUIREMENTSMET IT IS PURE LEG WORK IF STILL IN DIFFICULTY PLEASE COME BACK     A,A'    A,B'   A,C'    B,A'   B,B'    B,C'   C,A'    C,B'    C,C' A,A'    A,A'   A,B'    A,C'   B,A'    B,B'   B,C'    C,A'   C,B'    C,C' A,B'    A,B'                              A,C'    A,C'       B,A'    B,A'       B,B'    B,B'       B,C'    B,C'       C,A'    C,A'       C,B'    C,B'       C,C'    C,C'

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.