Question

In: Advanced Math

Prove that there is only one possible multiplication table for G if G has exactly 1,...

Prove that there is only one possible multiplication table for G if G has exactly 1, 2, or 3 elements. Analyze the possible multiplication tables for groups with exactly 4 elements, and show that there are two distinct tables, up to reordering the elements of G. Use these tables to prove that all groups with < 4 elements are commutative.

(You are welcome to analyze groups with 5 elements using the same technique, but you will soon know enough about groups to be able to avoid such brute-force approaches.)

Expert Solution

If G has exactly one element say then we must have the group identity and so the table is fixed at

*

If G has exactly two elements, one of them must be group identity so we must have

The multiplications must hold and

So we must have the group identity and so the group table is fixed at

*

If G has exactly 3 elements we must have

Then we must have

Consider the element . This can't be equal to since

This means we must have and similarly

And so our table is again fixed as:

*

Colby Messinger answered 2 years ago

prove by induction that for any simple connected graph G, if G has exactly one cycle...

prove by induction that for any simple connected graph G, if G has exactly one cycle then G has the same number of edges and nodes

Let G be a graph. prove G has a Eulerian trail if and only if G...

Let G be a graph. prove G has a Eulerian trail if and only if G has at most one non-trivial component and at most 2 vertices have odd degree

Prove that a subgroup H of a group G is normal if and only if gHg−1...

Prove that a subgroup H of a group G is normal if and only if gHg−1 =H for all g∈G

Prove that each point has exactly 1 polar in Desargues' configuration.

prove that f(x)=x^2019 +x-1 has only one real root

One possible performance enhancement is to do a shift and add instead of an actual multiplication....

One possible performance enhancement is to do a shift and add instead of an actual multiplication. Since 9 × 6, for example, can be written (2 × 2 × 2 + 1) × 6, we can calculate 9 × 6 by shifting 6 to the left 3 times and then adding 6 to that result. Show the best way to calculate 0x33×0x55 using shifts and adds . Assume both inputs are 16-bit unsigned integers

ii. Let G = (V, E) be a tree. Prove G has |V | − 1...

ii. Let G = (V, E) be a tree. Prove G has |V | − 1 edges using strong induction. Hint: In the inductive step, choose an edge (u, v) and partition the set vertices into two subtrees, those that are reachable from u without traversing (u, v) and those that are reachable from v without traversing (u, v). You will have to reason why these subtrees are distinct subgraphs of G. iii. What is the total degree of a...

1. Allow a one-time wave of immigrants, but only those who bring exactly the amount of...

1. Allow a one-time wave of immigrants, but only those who bring exactly the amount of capital needed to leave the steady-state capital/labor ratio unchanged. What would happen to per capital income in the short and long-run? Sketch the time path of k to show what happens in the short and longer term. 2. Allow a one-time wave of penny-less immigrants who bring no capital at all. What would happen to per capita income in the short and long run?...

Prove that: a) |sinx|<= |x| b) x = sin x has only one solution in real...

Prove that: a) |sinx|<= |x| b) x = sin x has only one solution in real number using mean value theorem

Prove that for n ⩾ 2 there are exactly two n-vertex graphs with n − 1...

Prove that for n ⩾ 2 there are exactly two n-vertex graphs with n − 1 distinct degrees (up to isomorphism). The other answers on the website are incorrect.