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Presentation of Groups

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• Richard H. Crowell 4 &
• Ralph H. Fox 5  

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 57))

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In this chapter we give a firm foundation to the concept of defining a group by generators and relations. This is an important step; for example, if one is not careful to distinguish between the elements of a group and the words that describe these elements, utter confusion is likely to ensue.

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There are a number of similar problems which are known to have no general solution: deciding whether or not the group defined by a given presentation is trivial (the triviality problem), is finite, is abelian, is free, etc; deciding whether or not a given word is a consequence of a given set of words (the word problem); and many others. See M. O. Rabin, “Recursive Unsolvability of Group Theoretic Problems,” Annals of Mathematics , Vol. 67 (1958), pp. 172–194.

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Department of Mathematics, Dartmouth College, Hanover, New Hampshire, 03755, USA

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Princeton University, Princeton, New Jersey, USA

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Crowell, R.H., Fox, R.H. (1963). Presentation of Groups. In: Introduction to Knot Theory. Graduate Texts in Mathematics, vol 57. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-9935-6_5

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Students, Faculty Expand Research on Reparations for JCSU

Charlotte, N.C./May 6, 2024 - “We are taking a stand on a crime against humanity.”

Is JCSU owed reparations? It’s a question a group on campus is posing in the form of an oral presentation and white paper, 109 pages long: “The HBCU Choice Bill: White Paper & Reparations Toolkit for JCSU Leadership.”

On Monday, May 6, 2024 Dr. Laurie Porter, Dr. Sabina Otienoburu, JCSU students Ashley Hall ’24 and Taylor Dobson ’24 spoke before a small group of JCSU administrators and staff. Each member of the group shared evidence in support of institutional reparations for the University.  

Reparations, can be defined as the making of amends through financial or other contributions to those who have been wronged. In the case of American history, reparations often refers to financial restitution owed Black Americans whose ancestors worked the land without compensation. Full reparations, according to the JCSU group “is an international law term that ‘indicates the degree in which reparations must redress crimes against humanity.’ Under International Law, reparations must ‘wipe out all consequences’ of a crime or harm.”

Reparations has been a topic of conversation since General Sherman’s special Field Order No. 15, issued January 16, 1865. The agreement would have allotted 400,000 acres of land to former slaves, a coastal plot stretching from South Carolina to Florida. The bill was overturned by President Andrew Jackson. Another bill introduced in the US Senate in 1894 would have granted each ex-slave $500 in reparations plus a monthly pension of $4 - $15. According to the NAACP, the bill died in congressional committee.

The group of JCSU students and faculty have been intrigued by the topic of reparations since Spring 2023. Their interest was piqued by the viewing of the documentary, “The Big Payback,” co- produced by actress Ericka Alexander and award-winning documentarian Whitney Dow. The film traces the mission of a young alderwoman in Illinois to pass the first tax-funded reparations for Black Americans in her town of Evanston.  

On this journey, JCSU students began to think about the history of Johnson C. Smith University, a history that dates back to 1867, a school built by the formerly enslaved. Prodded by the film, students began to ponder the inequities of the era, including education policies and segregation, North Carolina state tax code history, and the calculus of dollars spent/services rendered vs. services received and dollars owed: reparations.

The film, even in Dow’s eyes, was a conversation starter: “We like to think our film is a sort of activator.” The filmmaker hoped to instigate a re-interpretation of reparations through the lens of modern times.

“Most Americans look at reparations or even the idea of healing the rift between Black and white Americans as something that’s so big it’s almost impossible.”  

Ericka Alexander seemed committed to making the impossible, possible. She focused on community engagement as a brand of problem-solving: “Conversation is a natural thing, and it’s also how people can get to a solution.” For Alexander, the benefits of talking are even more far-reaching. “It’s also how people get educated.”

In March of 2023, Alexander and Dow held a reparations debate tour, passing through North Carolinas 10 HBCUs, including JCSU. Dr. Laurie Porter’s team of students argued a case for JCSU, presenting research in support of the HBCU Choice Bill. 

“The systematic denial of Black taxpayers to public education through the University of North Carolina education system created dramatic wealth and mobility gaps,” said Dr. Laurie Porter, Associate Professor of Communications.  

This led to the creation of private HBCUs like Johnson C. Smith University, where Black students were welcomed. While partnerships with the Presbyterian Church and later the James B. Duke Endowment and the United Negro College Fund (UNCF) helped to pay for students’ education in JCSU’s early years, the cost of a private education began to steeply rise in comparison to state-funded institutions.

“Private HBCUs are the least funded institutions across the board,” said Porter.  Now Porter, Otienoburu, Hall and Dobson are on a mission to fill the void.

During the May 6 session, Porter shared the group’s financial assessment. In 2023, for example, the average cost for attending a private HBCU was between $26,000 and $30,000 a year, while UNC Charlotte’s tuition costs hovered just over $7,000 on average, per year. Further, several HBCUs have noticed that predominately White institutions (PWIs) have duplicated programs previously unique to HBCUs.  

The affordability of public education has created a disruption in historical models of educational supply and demand. As a result, a number of students who were interested in a private HBCU experience have turned to the public education system for their post-secondary degrees. Nationwide, HBCU enrollment has dropped 16 percent, according to the National Center for Education Statistics. 

In order to rectify the issue, Porter, along with Otienoburu, Hall and Dobson propose the creation of an Urban Institute, created and run by JCSU, with existing and potential corporate and non-profit partners.

Hall and Dobson said they polled a number of students and alumni about reparations, and they were surprised to learn that their poll participants had little to no knowledge on what reparations were.

“Many older alumni we interviewed are against reparations because they feel the ship has sailed on that opportunity,” Hall said. “While our current students had little knowledge of reparations and the institution’s history, they have an immense pride for JCSU.”

After hearing their presentation, Kinloch showed her support for seeking reparations as an institution. 

“Reparations are important,” she said. “We are owed so much. How do we get what we deserve without being placed under a system that never extended their hand to help in the first place?”

Kinloch discussed several other solutions, such as continuing conversations with existing and potential partners as well as forming relationships with other local universities to expand academic programming and services. 

“Your presentation needs to be heard by others,” said Kinloch. 

Porter, Otienoburu, and JCSU students plan to continue to build on their research and present new findings to Dr. Kinloch and the JCSU Board of Trustees in the near future.

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