# Groups and matrices

1. Let G=GL(2,Q), Q meaning rational numbers, and let A =matrix 0 -1

1 0

and B =matrix 0 1

-1 1

Show that A^4 = I = B^6, but that (AB)^n does not equal I for all n >0. Conclude that AB can have infinite order even though both factors A and B have finite order( this cannot happen in a finite group)

I=identity

2.

Define W=[(1 2)(3 4)], the cyclic subgroup of symmetric group with n=4 (S sub 4) generated by (1 2)(3 4). show that W is a normal subgroup of V but that W is not a normal subgroup of S sub 4. Conclude that normality is not transitive: K is a normal subgroup of H and H is a normal subgroup of G need not imply K is a subgroup of G.

V= four group

I really dont know how to do these, I need to learn them in written proof form and I struggle.

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#### Solution Summary

This provides an example of a proof regarding a matrix with infinite order, and a proof regarding normal subgroups.

Boston Consulting Group Matrix

Can you help me complete this chart? We can use any 2 products from the same company. Thanks

1. Is the Boston Consulting Group (BCG) matrix reliable? Why/why not?

2. Given the shared template of BCG, choose any 2 products from the same company ( the below are examples only) and place them in the applicable quadrants of your choice. Explain the finding based on the applicable quadrant.

Cadbury Milk Chocolate Cadbury Yogurt Snacks (only example please do not use)

Stars Question Marks

Cash Cows Dogs

Part 1 - Table

Given this template of BCG, choose any 2 products and place them in the applicable quadrants of your choice. Explain the finding based on the applicable quadrant.

Cadbury Milk Chocolate

Stars

II Cadbury Yogurt Snacks

Question Marks

I

Cash Cows

III

Dogs