Mormons Not FLDS
The simple answer is ‘no’.
The historically informed and more complete answer is ‘yes, there is a historical connection’. Mormons (members of The Church of Jesus Christ of Latter-day Saints) do not share many of the beliefs of the Fundamentalist Church of Jesus Christ of Latter-day Saints (FLDS) Church, though there are a few that are shared.
Both churches believe in the concept of “continuing revelation” — that the heavens are not closed, and that prophets do in reality live today, in our time, and speak for God to all mankind. Both churches also believe that the Book of Mormon is another testament of Jesus Christ, and as such is scripture. This is why FLDS members still consider themselves “Mormons”, which the LDS Church disputes.
We differ, however, in belief concerning who holds the true priesthood of God on earth, each Church claiming that for themselves. The LDS Church no longer believes in the necessity of practicing polygamy, nor in the doctrine of adoption, nor in the practice of having all things in common, all of which the FLDS church continues to believe and practice.
It is clear that had not early LDS Church prophets such as Joseph Smith Jr., Brigham Young and John Taylor taught and practiced polygamy, that the FLDS church members would not be doing so today. The LDS Church stopped practicing polygamy between the years of 1890 and 1904, though it didn’t finally become fully eradicated as an LDS practice until the Presidency of Heber J. Grant, who served from 1918 to 1945 as Prophet of the Mormon Church.
Disaffected Mormons left (or were excommunicated from) the LDS Church and established the FLDS church primarily so they could continue the practice of polygamy.
Today, any LDS Mormons who are found to be practicing polygamy are excommunicated from the Mormon Church. I believe this is the basis for the simple statement that today the LDS church is not connected to the FLDS church.