Let G be a group of order 143 = 11 × 13, and, as usual, let Z(G) denote the center of G. Assume that we have found an element x ∈ Z(G) with x = e. What are the possibilities for the |Z(G)|? Prove any assertions you make.

Step-by-step explanation:

Let G be a group of order 143.

By the Sylow Third Theorem, the number of Sylow 11-subgroups of G is 1+11k and is a factor of 13,

Then, k=0 and there is an element a∈G such that Ord(a)=11 and ⟨a⟩ is normal to G.

Also, the number of Sylow 13-subgroups of G is 1+13k and is a factor of 11,

Then, k=0 and there is an element b∈G in such away that Ord(b)=13 and ⟨b⟩ is normal to G.

Since ⟨a⟩∩⟨b⟩={e}, it was discovered that ab=ba, therefore Ord(ab)=143.

the population of other specie of birds will decline or other specie of birds will find some alternate source of food to survive.

explanation:

according to the competitive exclusion principle, which is also known as gause's law, two specie living in the same niche and feeding on the same resource cannot co-exist stably.

as we have well studied the theory of natural selection which states that only those organisms survive which have better tendency to compete for food and adapt to the environmental conditions.

coming towards the question, what wi ll happen when one species outcompetes the other for the small fish , the other specie will not get enough food and will start to die or decline in population. eventually either other specie needs to find another alternative food source for itself or migrate at some other location where there is less competition otherwise it will just not exist anymore at that niche.

hope it !

you didn't post any answer choices, so its not 100% clear what your teacher is looking for

however, log(15/7) is equivalent to log(15) - log(7)

i'm using the rule log(x/y) = log(x) - log(y)