I would like to ask whether my basic understanding on statistics is true:

(1) After sampling from a population, the obtained "sample" which is a group of individuals "monitoring"/ "observation" has normal distribution and a standard deviation sqrt([(u-x)^2]/(n-1)).

(2) The "standard error of mean (SEM)" is the standard deviation of a set/group of "mean of sample", and this set of "mean of sample" shall have "t-distribution", and the standard deviation of this group (i.e. SEM) is just the standard deviation of the "t-distributed" set of "mean of sample"

(3) If taking infinity samples (with finite sample size), the distribution of the set/group of "mean of sample" will tend to be normal distribution.

If the above understanding is true, why the SEM is equal to SD of "one particular sample (set)" divided by square root n ?

Shall there be an assumption that the sample standard deviation is equal to the population standard deviation. However, is there any mathematics rationale for this assumption? If sampling is random, why the standard deviation of the sample can be assumed equal to population standard deviation?

Thank you very much

The standard deviation of a sample is